THE  PHASE  RULE 


WILDER   D.   BANCROFT 


The  Journal  of  Physical  Chemistry 
Ithaca,  New  York 

1897 


Get* 


PREFACE. 


THERE  has  been  so  much  work  done  in  physical  chemistry  dur- 
ing the  last  ten  years  that  the  mass  of  accumulated  material  is 
now  too  large  to  be  remembered  as  miscellaneous  facts.  It  becomes 
comparatively  easy  to  survey  the  whole  field  if  we  consider  the 
phenomena  as  examples  illustrating  a  few  general  principles.  Nat- 
ural divisions  of  the  subject  are  Qualitative  Equilibrium,  Quantita- 
tive Equilibrium,  Electrochemistry  and  Mathematical  Theory.  The 
last  two  divisions  are  already  recognized  as  sound  ;  but  the  qualita- 
tive and  quantitative  phenomena  have  always  been  treated  together 
and  the  distinction  between  them  has  nowhere  been  clearly  defined. 
My  idea  is  that  all  qualitative  experimental  data  should  be  presented 
as  particular  applications  of  the  Phase  Rule  and  the  Theorem  of 
Le  Chateh'er  while  the  guiding  principles  for  the  classification  of 
quantitative  phenomena  should  be  the  Mass  Law  and  the  Theorem 
of  van  't  Hoff. 

In  this  book  I  have  tried  to  present  the  subject  of  qualitative 
equilibrium  from  the  point  of  view  of  the  Phase  Rule  and  of  the 
Theorem  of  Le  Chatelier.  without  the  use  of  mathematics.  That 
such  a  treatment  should  be  possible  is  due  very  largely  to  the  work 
of  H.  W.  Bakhuis  Roozeboom  who  has  done  far  more  than  any  one 
else  to  show  the  importance  and  significance  of  Gibbs's  Phase  Rule. 


W.  D.  B. 


Cornell  University. 
February 


CONTENTS 


CHAPTER  I 

INTRODUCTION 

Phase  and  component,  i.  Phase  Rule,  2.  Theorem  of  Le  Chatelier,  4. 
Influence  of  gravity,  5.  Freedom  from  assumptions,  5. 

ONE  COMPONENT 

CHAPTER  II 

GENERAL  STATEMENT 

Solid,  liquid  and  vapor,  6.  Addition  of  work  and  heat,  8.  Changes  at  con- 
stant volume,  10.  Liquid  and  vapor,  ir.  Boiling-point,  12.  Critical  tempera- 
ture, 14,  Solid  and  vapor,  16.  Solid  and  liquid,  17.  Vapor,  19.  Liquid,  20. 
Solid,  21.  Labile  states,  22. 

CHAPTER   III 

WATER,    SULFUR   AND   PHOSPHORUS 

Water,  24.     Sulfur,  28.     Phosphorus,  32.     Allotropy,  33. 

TWO  COMPONENTS 

CHAPTER  IV 

ANHYDROUS  SALT  AND  WATER 

Compound  and  solution,  35.  Solvent  and  solute,  36.  Potassium  chloride 
and  water,  37.  Cryohydrates,  38.  Solubility  and  temperature,  41.  Critical 
temperature,  44.  Freezing-points,  46.  Freezing-mixtures,  47.  Divariant  sys- 
tems, 50. 

CHAPTER  V 

HYDRATED   SALTS 

Molecular  and  chemical  compounds,  56.  Sodium  sulfate  and  water,  37. 
Pressure  relations,  58.  Efflorescence,  59.  Heat  of  solution,  63.  Dissociation 
pressures,  65.  Solubility  curves,  66.  Supersaturation,  68.  Rational  diagram, 
70.  Pressure  curves  for  calcium  chloride  and  water,  71.  Solubility  curves  for 
calcium  chloride  and  water,  75.  Data,  77.  Ammonia  and  ammonium  bromide, 
78.  Solubility  curves  for  ferric  chloride  and  water,  79.  Stable  supersaturated 
solutions,  81.  Data,  83.  Thorium  sulfate  and  water,  85. 


vi  Contents. 

CHAPTER  VI 

VOLATILE  SOLUTES 

.  Iodine  and  chlorine,  86.  Solubility  curves,  88.  Data,  90.  Gases  and 
liquids,  91.  Volatilization  of  solids,  92. 

CHAPTER  VII 

TWO    LIQUID    PHASES 

Naphthalene  and  water,  94.  Types  of  partially  miscible  liquids,  96.  Equi- 
librium between  partially  tniscible  liquids,  99.  Boiling-points.  101.  Solubility 
and  temperature,  102.  Consolute  temperature,  103.  Phenol,  benzoic  acid  and 
salicylic  acid  with  water,  105.  Triethylamine  and  water,  To6.  Sulfur  dioxide 
and  water,  107.  Concentration  temperature  diagram,  1 10.  Data,  in.  Hydro- 
bromic  acid  and  water,  1 12.  Data,  114. 

CHAPTER  VIII 
CONSOLUTE   LIQUIDS 

Naphthalene  and  phenanthrene,  116.  Eutectic  mixture,  117.  Vapor  pres- 
sure of  consolute  liquids,  118.  Boiling-points,  120.  Fractional  distillation,  124. 
Solubility  of  consolute  liquids,  126.  Fusion  and  solubility  curves,  128.  Melted 
salts,  130.  Solubility  curves  when  compounds  are  formed,  131.  Rule  for  num- 
ber of  constant  freezing-points,  132.  Two  modifications  of  the  solvent,  133. 
Mercuric  and  silver  iodides,  134. 

CHAPTER  IX 

SOLID   SOLUTIONS 

Freezing-points,  135.  Solid  and  liquid  solutions  with  the  same  composi- 
tion, 137.  Two  sets  of  solid  solutions,  137.  Occlusion  and  adsorption,  139. 
Palladium  and  hydrogen,  139.  Occlusion  of  gases  by  carbon,  140. 

CHAPTER  X 
REVIEW 

Typical  nonvariant  systems,  141.  Alloys,  142.  Metallic  compounds,  143. 
Metallic  solid  solutions,  144.  Amorphous  antimony,  145. 

THREE  COMPONENTS 

CHAPTER  X[ 

GENERAL  THEORY 

Graphical  methods,  146.  Triangular  diagram,  147.  Simplest  cases,  148. 
Theorem  of  van  Rijn  van  Alkemade,  149.  One  compound,  AC,  150.  Two  com- 
pounds, AC  and  AC,  152.  Two  compounds,  AC  and  ABC,  153.  Two  com- 


Contents.  vii 

pounds,  AC  and  BC,  153.  Potassium,  sodium  and  lead  nitrates,  154.  Iso- 
therms, 155.  Two  subdivisions,  156.  Potassium  chloride,  potassium  nitrate 
and  water,  157.  Disappearance  of  water.  158.  "Break"  when  the  solvent 
changes,  159.  General  form  of  isotherms  when  compounds  can  exist,  161. 

CHAPTER  XII 
TWO  SALTS  AXD  WATER 

Potassium  sal  fate,  magnesium  snlfate  and  water,  165.  Abnormal  cryohy- 
dric  temperature,  166.  Behavior  of  the  double  salts,  167.  Rule  of  Meyerhof- 
fer,  169.  Diagram  of  van  der  Heide,  170.  Application  of  the  theorem  of  van 
Rijn  van  Alkemade,  170.  Isotherms,  171.  Data,  171.  Sodium  snlfate,  mag- 
nesium snlfate  and  water,  172.  Overlapping  fields,  173.  Range  of  decomposi- 
tion, 174.  Copper  chloride,  potassium  chloride  and  water,  175.  Stability  of 
double  salt,  177.  Rule  for  cryohydric  points,  1 78.  Potassium  iodide.  lead  iodide 
and  water,  179.  Calcium  acetate,  copper  acetate  and  water,  179.  Maxininm 
and  minimum  temperatures  for  double  salts.  180.  Reason  for  the  rule,  183. 

CHAPTER  XIII 
PRESSURE  CURVES 

Sodium  sulfate.  sodium  chloride  and  water,  184.  Efflorescence  under  water. 
186.  Solubility  of  hydrated  salt,  187.  Calcium  acetate,  copper  acetate  and 
water,  188.  Sodium  snlfate,  magnesium  snlfate  and  water,  190.  Combined 
diagram,  192.  Copper  chloride,  potassium  chloride  and  water,  194.  Efflores- 
cence of  hydrated  double  salts,  195.  Impossible  formula  for  a  hydrated  double 
salt,  195.  Behavior  of  lead  potassium  iodide,  196.  Intersection  of  dissociation 
curves,  197.  Ammonium  chloride  and  lead  oxide,  198. 

CHAPTER  XIV 
SOLID  SOLUTIONS 

Ether  and  rubber,  199.  Starch  and  iodine,  199.  Dyeing  of  silk,  200.  Oc- 
clusion by  charcoal,  200. 

CHAPTER  XV 
ISOTHERMS 

Continuous  addition  of  one  component,  201.  Salts  do  not  react.  202.  Sin- 
gle series  of  solid  solutions,  203.  Two  series  of  solid  solutions,  204.  One 
double  salt,  204.  Two  double  salts,  205.  One  double  salt  and  one  series  of 
solid  solutions,  206.  One  double  salt  and  two  series  of  solid  solutions,  207. 
Three  series  of  solid  solutions,  207.  General  rule,  208. 


viii  Contents. 

CHAPTER  XVI 

FRACTIONAL   EVAPORATION 

Isothermal  evaporation  with  removal  of  crystals,  209.  Salts  do  not  react, 
210.  Single  series  of  solid  solutions,  210.  Two  series  of  solid  solutions,  212. 
One  double  salt,  212.  Behavior  of  lead  potassium  iodide,  213.  Two  double 
salts,  215.  One  double  salt  and  one  series  of  solid  solutions,  216.  One  double 
salt  and  two  series  of  solid  solutions,  217.  Three  series  of  solid  solutions,  218. 

CHAPTER  XVII 

TWO  VOLATILE   COMPONENTS 

Ferric  chloride,  hydrochloric  acid  and  water,  219.  Hydrated  double  salts, 
221.  Application  of  the  theorem  of  van  Rijn  van  Alkemade,  222.  Forms  of 
the  isotherms,  223.  Data,  224. 

CHAPTER  XVIII 

COMPONENTS    AND   CONSTITUENTS 

Dependent  and  independent  variables,  226.  Limiting  conditions,  227. 
Water  as  monobasic  and  dibasic  acid,  229.  Adding  degrees  of  freedom,  230. 
Classification  independent  of  the  chemical  elements,  231.  Systems  with  meta- 
thetical  reactions,  231.  Barium  sulfate  and  sodium  carbonate,  232.  Mercuric 
sulfate  and  water,  232.  Tartrates  and  racemates,  233.  Passive  resistance  and 
time,  234.  Semipermeable  diaphragms,  235.  Treatment  of  osmotic  pressure, 
236.  Two  different  diaphragms,  236.  Adiabatic  diaphragms,  237. 

CHAPTER  XIX 

TWO    LIQUID   PHASES 

Second  liquid  phase  independent  of  the  third  component,  238.  Three  liquid 
solutions,  239.  Break  in  the  isotherm.  240.  Second  liquid  phase  dependent  on 
the  third  component,  240.  Ammonium  sulfate,  alcohol  and  water,  241.  Labile 
equilibrium,  242. 

FOUR  COMPONENTS 
CHAPTER  XX 

GENERAL  THEORY 

Metathetical  reactions,  243.  Two  pairs  of  salts  and  water,  243.  Stable  and 
instable  pairs,  244.  Number  of  components,  244.  Data  for  inversion  points, 
245.  Magnesium  sulfate,  potassium  chloride  and  water,  246.  Effect  of  adding 
magnesium  chloride,  246.  Data,  247.  Pressure  relations  when  the  two  mag- 
nesium sulfates  are  solid  phases,  247.  Indirect  evidence,  248. 


CHAPTER  I 

INTRODUCTION 

The  two  expressions  describing  in  a  qualitative  manner  all  states 
and  changes  of  equilibrium  are  the  Phase  Rule  and  the  Theorem  of 
Le  Chatelier.  A  phase  is  defined  as  a  mass  chemically  and  physi- 
cally homogeneous1  or  as  a  mass  of  uniform  concentration,  the  num- 
ber of  phases  in  a  system  being  the  number  of  different  homogeneous 
masses  or  the  number  of  masses  of  different  concentration.  In  the 
case  of  water  in  equilibrium  with  its  own  vapor  there  is  the  liquid 
and  the  vapor  phase,  two  in  number.  If  there  is  a  salt  dissolved  in 
the  water  there  are  still  two  phases,  the  liquid  or  solution  phase  and 
the  vapor  phase.  If  ice  crystallizes,  there  is  added  a  solid  phase  and 
the  number  becomes  three.  If,  in  addition,  the  dissolved  substance 
separates  in  the  solid  form  or  as  a  second  liquid  layer,  there  will  be 
four  phases  present,  the  vapor,  liquid  and  two  solid  phases  or  the  va- 
por, solid  and  two  liquid  phases  as  the  case  may  be.  Although  the 
ice  separates  in  many  crystals,  yet  each  is  like  every  other  in  compo- 
sition and  density  and  taken  together  they  constitute  one  phase.  If 
the  crystals  were  not  alike  as  is  the  case  with  rhombic  and  inonoclinic 
sulfur  they  would  form  as  many  phases  as  there  were  kinds  of  crys- 
tals, two  in  the  example  just  cited,  three  if  we  have  diamond,  graph- 
ite and  carbon.  The  components  of  a  phase  or  system  are  defined 
as  the  substances  of  independently  variable  concentration  in  the 
phase  or  system  under  consideration.  A  component  need  not  be  a 
chemical  compound,  that  is  a  substance  described  by  the  Theorem  of 
Definite, and  Multiple  Proportions,  though  this  is  usually  the  case. 
For  instance,  a  mixture  of  propyl  alcohol  and  water  in  such  propor- 
tions that  the  percentage  composition  of  the  liquid  is  the  same  as  that 
of  the  vapor  might  be  treated  as  one  component  ;  but  there  is  no  ad- 
vantage in  this,  as  it  is  true  for  only  one  temperature  and  when 


1  Cf.  Gibbs,  Trans.  Conn.  Acad.  3,  152  (1876). 


2  The  Phase  Rule 

there  are  no  other  components.  The  main  point  to  be  observed  in 
determining  the  number  of  components  in  a  given  system  is  that 
each  compound  is  not  necessarily  a  component.  Thus  a  hydrated 
salt  is  to  be  treated,  when  in  equilibrium  with  the  solution  or  vapor, 
as  made  up  of  salt  and  water  and  is  not  in  itself  a  component.  The 
same  holds  true  of  a  double  salt  such  as  the  double  sulfates  of  cop- 
per and  potassium.  Here  the  components  are  the  two  single  salts 
and  water  because  the  concentration  of  these  three  can  be  varied 
and  they  are  sufficient  to  form  all  modifications  which  can  exist.  If 
one  is  treating  calcium  carbonate  in  equilibrium  with  calcium  oxide 
and  carbonic  acid,  there  are  only  two  components,  calcium  oxide  and 
carbonic  acid  ;  for  the  calcium  carbonate  is  merely  a  solid  phase  con- 
taining the  two  components.  The  fact  that  the  two  components 
unite  to  form  a  phase  in  definite  proportions  does  not  have  anything 
to  do  with  the  matter.  On  the  other  hand  it  is  not  permissible  to 
take  calcium  and  oxygen  as  two  of  the  actual  components  of  this 
system  because  they  are  neither  independent  variables  nor  are  they 
in  equilibrium  with  the  system.1 

It  has  been  shown  by  Gibbs'that  the  state  of  a  phase  is  completely 
determined  if  the  pressure  and  temperature  together  with  the 
chemical  potentials  of  its  components  be  known.  There  is  therefore 
an  equation  connecting  these  quantities  which  will  describe  the 
phase.  For  each  other  phase  in  equilibrium  with  the  first  there  will 
be  another  equation  containing  the  same  variables. 

There  will  thus  be  the  same  number  of  equations  as  there  are 
phases,  while  the  number  of  independent  variables  will  equal  the 
number  of  components  plus  the  temperature  and  pressure.  If  the 
number  of  components  be  "«"  the  number  of  variables  will  be 
"  n  +  2."  This  is  true  only  in  case  we  are  considering  a  system 
uninfluenced  by  gravity,  electricity,  distortion  of  the  solid  masses  or 
capillary  tensions  because  it  is  only  when  the  effects  due  to  these  in- 
fluences are  removed  that  the  values  for  the  pressure,  temperature 
and  chemical  potentials  are  uniform  throughout  the  whole  system. 
While  we  do  not  know  the  single  equations  referred  to  nor  the 
chemical  potentials  of  the  components,  it  is  possible  to  draw  some 

1  Nernst,  Theor.  Chem.  482. 

2  Trans.  Conn.  Acad.  3,  152  (1876). 


Introduction  3 

conclusions  in  respect  to  the  possible  number  of  states  of  equilibrium 
in  any  given  case.  Since  the  number  of  independent  variables  is 
always  equal  to  "  n  +  2  "  by  definition  and  the  number  of  equations 
equals  the  number  of  phases,  it  follows  that  in  a  system  of  "  n  +  2  " 
phases  there  will  be  as  many  theoretical  equations  as  there  are 
variables ;  in  other  words,  that  each  of  the  variables  has  one 
value  and  one  only  for  a  given  set  of  "«  +  2"  phases.  A 
given  combination  of  "  n  +  2"  phases  can  exist  at  one  temper- 
ature and  one  pressure  only,  the  composition  of  the  phases  being  also 
definitely  determined.  Such  a  system  is  called  a  non variant  system , 
the  temperature  and  pressure  at  which  alone  it  can  exist  are  known 
as  the  inversion  temperature  and  pressure.1  If  there  are  only 
"'  n  +  i "  phases,  the  system  is  no  longer  completely  defined  and  has 
one  degree  of  freedom.  It  is  therefore  called  a  monovariant  system. 
If  we  fix  arbitrarily  one  of  the  variables,  say  the  pressure  or  the 
temperature,  the  system  is  again  entirely  defined.  The  character- 
istics of  the  monovariant  system  are  that  for  a  given  combination  of 
phases  there  is  for  each  temperature  one  pressure  and  one  set  of  con- 
centrations for  which  the  system  is  in  equilibrium  ;  for  each  pressure, 
one  temperature  and  one  set  of  concentrations  ;  for  each  set  of  con- 
centrations, one  pressure  and  one  temperature.  A  system  composed 
of  "»"  phases  is  called  a  di variant  system.  In  it  there  are  two 
variables  which  can  be  fixed  arbitrarily  before  the  system  is  com- 
pletely defined.  In  such  a  system,  for  a  given  temperature,  it  is 
possible  to  have  a  series  of  pressures  by  changing  the  concentrations 
or  a  series  of  concentrations  by  changing  the  pressures.  For  a  given 
pressure  the  temperatures  can  vary  with  changing  concentrations 
and  -oue--!jersa  while  for  definite  concentrations  there  are  similar  rela- 
tions between  the  pressures  and  temperatures.  If  instead  of  "  n lf 
phases  the  system  contains  n  —  i,  n  —  2,  etc. ,  phases  it  is  known  as 
a  trivariant,  tetra variant  system,  etc.  There  are  other  terms  in  use, 
a  monovariant  system  being  called  a  * '  case  of  complete  heterogeneous 
equilibrium  "  while  a  di  variant  system  is  known  as  a  case  of  "  in- 
complete heterogeneous  equilibrium. ' '  *  These  phrases  are  unwieldy 


1van  't  Hoff.  Etudes  142. 

'Bakhuis  Roozeboom,  Recueil  Trav.  Pays-Bas  6,  266  (1887). 


4  The  Phase  Rule 

and  unsatisfactory  and  must  give  way  to  the  more  rational  nomencla- 
ture adopted  here.1 

By  increasing  the  number  of  components  and  decreasing  the  num- 
ber of  phases  it  is  possible  to  make  a  system  with  almost  any  degree 
of  freedom  ;  but,  practically,  a  system  ceases  to  be  interesting  from 
the  qualitative  point  of  view  when  it  contains  less  than  "  n  "  phases 
because  the  possibilities  are  so  numerous  and  so  ill-defined.  In  the 
other  direction,  that  of  decreasing  the  components  and  increasing 
the  phases,  it  is  impossible  to  go.s  Since  "  n  -\-  2"  phases  consti- 
tute a  nonvariant  system  which  can  be  in  equilibrium  at  one  tem- 
perature and  pressure  only,  a  system  of  "n  +  j"  phases  is  most 
improbable  and  none  such  are  known  where  there  are  no  so-called 
passive  resistances  to  change.5  The  discussion  will  therefore  be 
limited  to  nonvariant,  monovariant  and  di variant  systems,  starting 
with  the  number  of  components  equal  to  one  and  increasing  to  four. 
Before  beginning  the  study  of  the  possible  variations  in  equilibrium 
caused  by  changing  the  different  variables  and  the  number  of  phases, 
it  is  necessary  to  have  some  clue  as  to  the  direction  of  the  change  in 
equilibrium  when  there  is  an  alteration  in  the  system.  This  is  given 
by  the  Theorem  of  I^e  Chatelier,  which  says  :  ' '  Any  change  in  the 
factors  of  equilibrium  from  outside  is  followed  by  a  reverse  change 
within  the  system."  *  If  the  external  pressure  is  raised  there  is  an 
increased  formation  of  the  component  or  phase  occupying  the  lesser 
volume ;  if  heat  is  added  there  is  increased  formation  of  the  compo- 
nent or  phase  involving  an  absorption  of  heat ;  if  the  concentration 
of  one  component  is  increased  in  a  given  phase  there  is  formation  of 
the  component  or  phase  which  involves  a  decrease  in  the  concentra- 
tion of  the  first  component.  In  other  words,  the  system  in  equili- 
brium tends  to  return  to  equilibrium  by  elimination  of  the  disturb- 
ing element.  It  is  now  possible  to  take  up  distinct  cases  and  see  the 


1  This  classification  of  systems  into  nonvariant,  monovariant,  di  variant  and 
so  on,  is  due  to  Professor  Trevor,  who  has  used  it  for  several  years  in  his  lec- 
tures. 

'Gibbs,  Trans.  Conn.  Acad.  3,  153  (1876). 
.   3Gibbs,  Trans.  Conn.  Acad.  3,  in  (1876). 

4  Comptes  reudus,  99,  786  (1884);  Braun,  Wied.  Ann.  33,  337  (1888)  ;  Cf. 
Duheni,  M£canique  chimique,  152. 


Introduction  5 

way  in  which  the  Phase  Rule  and  the  Theorem  of  Le  Chatelier  ap- 
ply.  It  must  be  borne  in  mind  that  we  are  discussing  only  the 
states  and  changes  of  equilibrium  which  are  due  to  the  pressure, 
temperature  and  concentrations,  and  that  the  disturbing  effects  due 
to  gravity,  electricity,  distortion  of  the  solid  masses  and  capillary 
tensions  are  eliminated.  If  this  is  the  case  the  problem  becomes 
simplified  since  the  absolute  mass  of  the  phase  has  no  effect  on  the 
equilibrium,  because  the  concentration  of  a  phase  is  not  a  function 
of  its  mass.  A  saturated  solution  remains  saturated  whether  it  is  in 
contact  with  a  small  or  a  large  amoifnt  of  the  solid.  In  the  same 
way  the  equilibrium  is  not  disturbed  if  the  bulk  of  the  solution  be 
poured  off.  This  would  not  be  true  if  we  were  taking  into  account 
the  effect  due  to  gravity.  Crystals  at  the  bottom  of  a  long  tube  filled 
with  solution  are  under  a  greater  pressure  than  if  the  liquid  layer 
were  but  a  few  millimeters  thick  -  and  have  different  solubilities  in 
the  two  cases.  This  is  verj-  noticeable  in  divariant  systems  of  three 
components  when  there  is  a  vapor  phase  in  equilibrium  with  two 
liquid  phases.  An  increase  in  the  amount  of  the  upper  liquid  layer 
produces  a  very  distinct  change  in  the  mass  of  the  lower  liquid 
phase.  This  is  a  point  which  has  been  completely  overlooked  in  the 
development  of  Nernst's  Distribution  Theorem.1  The  pressure  of  a 
gas  in  a  tall  cylinder  is  not  strictly  uniform  owing  to  the  influence  of 
gravity.  These  effects  as  a  rule  are  very  small  and  may  be  neg- 
lected in  most  cases  without  danger.  They  can  be  reduced  to  a 
minimum  by  working  with  small  quantities. 

It  will  be  noticed  further  that  the  classification  of  equilibria  under 
the  Phase  Rule  and  of  changes  of  equilibria  under  the  Theorem  of 
Le  Chatelier  is  perfectly  general  and  involves  no  assumptions  as  to 
the  nature  of  matter  or  of  the  changes  taking  place.  There  is  no 
need  of  assuming  that  matter  is  made  up  of  discrete  particles  nor 
that  it  is  continuous ;  there  is  even  no  need  of  assuming  its  exist- 
ence or  non-existence.  It  is  immaterial  whether  there  is  or  is  not  a 
distinction  between  "chemical"  and  "  physical  "  reactions.  It  is 
simply  a  question  of  the  relative  number  of  independently  variable 
components  and  phases  in  the  one  case  and  of  the  experimental  data 
in  regard  to  heat  effects,  densities  and  concentrations  in  the  other. 


Zeit.  phys.  Chem.  8,  110(1891). 


ONE    COMPONENT 
CHAPTER  II 

GENERAL    STATEMENT 

The  most  familiar  example  of  a  nonvariant  system  made  up  of 
one  component  is  the  equilibrium  between  solid,  liquid  and  vapor, 
as  in  the  system  composed  of  ice,  water  and  water  vapor,  or  solid, 
melted  and  vaporized  naphthalene.  The  application  of  the  Phase 
Rule  leads  us  to  expect  that  a  system  of  this  type  can  be  in  equili- 
brium at  only  one  temperature  and  one  pressure.  This  is  true  ex- 
perimentally, the  temperature  for  water  being  about  zero  degrees 
Centigrade 1  and  the  pressure  about  four  and  a  half  millimeters  of 
mercury.  In  the  ordinary  determinations  of  the  inversion  tempera: 
ture  in  open  vessels  there  are  really  two  components  instead  of  one, 
the  substance  under  consideration  and  air,  so  that  the  values  cited 
for  water  are  not  the  temperature  and  pressure  at  which  the  three 
modifications  of  water  would  be  in  equilibrium  if  they  alone  were 
present.  The  effect  of  the  air  in  changing  the  inversion  temperature 
is  due  to  its  solubility  and  also  to  its  pressure  upon  the  solid  and 
liquid  phases.  Since  the  effect  of  a  pressure  of  one  atmosphere  is 
very  slight  and  since  the  amount  of  air  dissolved  in  water  is  not 
usually  very  great,  the  value  of  the  inversion  temperature  as  found 
in  open  vessels  differs  only  slightly  from  the  true  value  and  for  pur- 
poses of  discussion  the  two  may  be  considered  identical  in  most 
cases.  This  is  especially  true  if  the  determination  is  made  by  melt- 
ing the  solid  in  presence  of  its  own  vapor  instead  of  freezing  the 
liquid.  In  Table  I,  are  the  inversion  temperatures  and  pressures  of 
several  substances.  The  first  column  of  figures  gives  the  tempera- 
ture in  Centigrade  degrees  ;  the  second,  the  pressure  in  millimeters 
of  mercury  : 


1  More  accurately  +  0.0066°'     Gossens,  Arch.  Neerl.  2O,  449  (1886). 


One  Component 

TABLE  I 

Bromine 

-7-o° 

-44-5 

Ice 

o.o 

4-6 

Benzene 

5-3 

.    35-4 

Acetic  acid 

16.4 

9-4 

Naphthalene 

79.2 

9- 

Iodine 

114.2 

90. 

Camphor 

175- 

354- 

If  westartfrom  any  system  in  equilibrium  it  is  possible,  b\-  adding 
or  subtracting  work  or  heat,  to  bring  about  changes  in  the  relative 
masses  of  the  phases  and,  under  suitable  conditions,  the  disappear- 
ance of  one  or  more  phases.  The  direction  of  these  changes  can  be 
predicted  from  the  Theorem  of  Le  Chatelier  if  we  know  the  densities 
and  concentrations  of  the  different  phases  and  the  sign  of  the  heat 
effect  when  one  phase  increases  at  the  expense  of  one  of  the  others. 
The  vapor  of  a  substance  is  less  dense  than  the  liquid  or  solid  modi- 
fication at  any  temperature  at  which  it  can  ba  in  equilibrium  with 
either  of  these.1  The  vapor  of  a  substance  is  less  dense  than  the 
liquid  or  solid  modification.  Increase  of  external  pressure  means 
therefore  decrease  of  the  vapor  phase  and  -Dice-versa.  Some  sub- 
stances are  more  and  some  are  less  dense  in  the  solid  than  in  the 
liquid  state  so  that  it  is  necessary  to  know  the  peculiarities  of  the 
system  under  consideration  in  order  to  tell  which  of  these  two  phases 
is  the  more  stable  under  increased  pressure.  The  change  of  solid 
into  liquid  and  of  liquid  into  vapor  alwa}Ts  involves  absorption  of  heat. 

These  preliminary  experimental  data  being  given  we  can  now  take 
up  the  changes  in  the  relative  masses  of  a  non variant  system,  solid, 
liquid  and  vapor.  If  we  have  this  system  in  a  vessel  closed  by  a 
movable  piston  so  that  it  is  possible  to  vary  the  pressure  and  volume 
we  shall  find,  if  we  keep  the  system  at  the  inversion  temperature, 
and  increase  the  external  pressure  that,  as  predicted  by  the  Theorem 
of  Le  Chatelier,  there  will  be  formation  of  a  system  occupying  a 
lesser  volume.  Some  of  the  vapor  will  condense  until  the  original 
pressure  is  restored,  the  volume  of  the  vapor  phase  decreasing,  that 


1  This  would  not  be  true  for  supercooled  vapors  near  the  critical  temperature  ; 
but  it  holds  for  all  cases  of  stable  equilibrium. 


8  The  Phase  Rule 

of  the  liquid  phase  increasing,  while  the  solid  phase  remains  con- 
stant. 

If  the  external  pressure  on  the  piston  be  kept  continuously  greater 
than  the  equilibrium  pressure  of  the  system,  the  condensation  will 
continue  until  the  vapor  phase  has  disappeared  and  there  is  present 
the  monovariant  system,  solid  and  liquid.  If  the  external  pressure 
be  less  than  the  equilibrium  pressure,  more  vapor  will  form  and  the 
vapor  phase  will  increase  in  volume  at  the  expense  of  the  liquid, 
driving  back  the  piston  till  the  equilibrium  pressure  is  reached.  If 
the  external  pressure  be  kept  continuously  less  than  the  equilibrium 
pressure,  the  evaporation  and  expansion  will  go  on  until  the  liquid 
has  disappeared  and  there  is  again  a  monovariant  system,  this  time, 
solid  and  vapor.  It  might  be  thought  that  by  bringing  the  solid  and 
liquid  phases  into  the  two  arms  of  a  £/-tube  so  that  each  is  in  contact 
with  the  vapor,  evaporation  would  take  place  from  both  surfaces.  In 
this  case,  by  taking  suitable  proportions  of  solid  and  liquid, 'it  would 
be  possible  to  make  the  former  disappear  before  the  latter  leaving 
the  monovariant  system,  liquid  and  vapor.  This  will  not  happen, 
practically,  because  the  conditions  of  the  experiment  are  impossible. 
Owing  to  surface  tension,  the  solid  will  be  completely  wetted  b}-  the 
liquid  and  will  not  be  in  direct  contact  with  the  vapor.  This  is  a 
complication  which  has  been  ruled  out  expressly  and  we  may  con- 
clude that,  apart  from  the  disturbing  influences  due  to  surface  ten- 
sion, it  would  be  possible  to  pass  by  change  of  external  pressure 
from  the  nonvariant  system,  solid,  liquid  and  vapor  to  the  mono- 
variant  system,  liquid  and  vapor.  In  the  purely  theoretical  case 
where  the  effect  of  surface  tension  is  eliminated,  it  would  be  possible 
to  have  increase  of  both  solid  and  liquid  at  the  expense  of  the  vapor 
phase  when  the  external  pressure  is  greater  than  the  equilibrium 
pressure. 

Instead  of  changing  the  external  pressure,  which  is  equivalent  to 
adding  work  to  or  taking  it  from  the  system,  it  is  possible  to  cause 
changes  in  equilibrium  by  adding  or  subtracting  heat.  This  is  done 
by  bringing  the  system  into  contact  with  a  body  at  another  tempera- 
ture than  its  own.  For  purposes  of  reference  this  outside  body  will 
be  called  the  heat  reservoir,  and  it  may  be  at  a  higher  or  lower  tem- 
perature than  the  system.  Heat  may  be  added  or  subtracted  while 


One  Component  9 

the  system  is  kept  at  constant  pressure  or  at  constant  volume,  the 
changes  in  the  two  cases  being  usually  different.  All  instances 
where  the  pressure  and  volume  change  simultaneous!}-  maj-  be  re- 
solved into  the  sum  of  two  changes,  one  at  constant  pressure,  the 
other  at  constant  volume.  So  long  as  we  have  a  nonvariant  system 
before  us,  the  addition  of  heat  can  bring  about  no  change  of  tem- 
perature and  it  is  necessary  only  to  consider  the  changes  in  the  rela- 
tive and  absolute  masses  of  the  phases.  If  the  system  is  at  a  lower 
temperature  than  the  heat  reservoir  and  is  kept  at  constant  pressure, 
the  solid  will  melt  partially  with  absorption  of  heat,  there  being  an 
increase  in  the  liquid  at  the  expense  of  the  solid  phase,  the  vapor 
phase  remaining  constant.  This  change  will  go  on  until  the  heat 
reservoir  becomes  of  the  same  temperature  as  the  system,  that  is  un- 
til no  more  heat  is  added.  If  the  heat  reservoir  be  kept  continuous- 
ly at  a  higher  temperature  than  the  system,  the  change  will  continue 
until  there  is  present  the  monovariant  system,  liquid  and  vapor. 
If  the  system  is  at  a  higher  temperature  than  the  heat  reservoir  and 
is  kept  at  constant  pressure,  the  reverse  change  will  take  place,  the 
solid  phase  increasing  at  the  expense  of  the  liquid  phase,  the  amount 
of  vapor  remaining  constant.  If  the  heat  reservoir  be  kept  continu- 
ously at  a  lower  temperature  than  the  system  there  will  be  formed 
eventually  the  monovariant  system,  solid  and  vapor.  The  change 
from  liquid  to  solid  is  accompanied  by  an  evolution  of  heat  and  is  in 
accordance  with  the  Theorem  of  Le  Chatelier.  It  is  to  be  noticed 
that  the  other  monovariant  system,  solid  and  liquid,  is  not  formed. 
It  is  not  possible  to  pass  at  constant  pressure  from  the  uonvariant 
system,  solid,  liquid  and  vapor  to  the  monovariant  system,  solid  and 
liquid  by  addition  or  subtraction  of  heat. 

If  the  nonvariant  system  be  kept  at  constant  volume  and  heat 
added  continually,  the  changes  and  the  resulting  monovariant  sys- 
tem will  depend  on  the  relative  densities  of  the  solid  and  liquid,  and 
the  relative  masses  of  the  three  phases.  If  the  solid  is  denser  than 
the  liquid — the  usual  case — the  volume  occupied  by  the  liquid  and 
solid  will  become  larger  as  the  solid  melts  and  the  volume  of  the 
vapor  will  become  less.  Under  these  circumstances  there  is  increase 
of  the  liquid  phase  at  the  expense  of  the  other  two.  If  the  quan- 
tity of  solid  is  large  and  that  of  the  vapor  very  small,  the  vapor 


io  The  Phase  Rule 

phase  will  disappear  first  and  the  resulting  monovariant  system  will 
be  composed  of  a  solid  and  a  liquid  phase.  If  the  amount  of  solid 
is  small  and  the  volume  of  the  vapor  relatively  large,  the  solid  phase 
will  be  the  first  to  vanish,  leaving  the  monovariant  .system,  liquid 
and  vapor.  If  the  solid  is  less  dense  than  the  liquid,  as  in  the  case 
of  ice  and  water,  the  total  volume  of  the  solid  and  liquid  phases 
will  decrease  with  the  melting  of  the  solid.  The  liquid  and  vapor 
phases  will  both  increase  at  the  expense  of  the  solid  phase,  and  the 
resulting  monovariant  system  will  be  made  up  of  liquid  and  vapor 
irrespective  of  the  original  masses  of  the  three  phases.  If  the  sys- 
tem is  brought  into  contact  with  a  heat  reservoir  kept  continually  at  a 
lower  temperature  than  its  own,  the  reverse  changes  will  take  place. 
If  the  solid  is  more  dense  than  the  liquid,  the  solid  and  vapor  phases 
will  increase  at  the  expense  of  the  liquid  phase,  forming  a  system  com- 
posed of  a  solid  and  a  vapor  phase  regardless  of  the  original  volumes 
of  the  three  phases.  If  the  solid  is  less  dense  than  the  liquid,  the 
solid  phase  will  increase  at  the  expense  of  the  liquid  and  vapor 
phases,  the  resulting  system  being  solid  and  vapor  or  solid  and 
liquid  as  the  volume  of  the  vapor  phase  is  large  or  small  relatively 
to  that  of  the  liquid.  It  must  be  kept  in  mind  that  the  main  change 
in  adding  or  subtracting  heat  is  the  conversion  of  solid  into  liquid 
and  vice  versa,  and  that  the  change  in  the  vapor  phase  is  a  second- 
ary one  due  to  the  difference  in  density  of  the  solid  and  liquid.  It 
is  evident  that  the  change  to  liquid  and  vapor  on  adding  heat  and  to 
solid  and  vapor  on  subtracting  it,  is  in  accordance  with  the  Theorem 
of  I/e  Chatelier,  the  formation  of  liquid  from  solid  being  accom- 
panied by  absorption,  the  formation  of  solid  from  liquid  by  evolu- 
tion of  heat.  This  is  not  so  obvious  when  the  final  state  is  solid 
and  liquid  on  addition  of  heat.  It  is  true  that  the  conversion  of 
vapor  into  liquid  is  accompanied  by  an  evolution  of  heat,  but  this 
change  is  secondary,  as  has  just  been  pointed  out.  There  is  very 
little  vapor  condensed  and  the  sign  of  the  heat  effect  is  determined 
by  the  much  larger  change,  as  far  as  mass  is  concerned,  of  solid  into 
liquid  ;  there  is  actually  an  absorption  of  heat  and  the  Theorem  of 
Le  Chatelier  is  again  confirmed. 

Having  considered  the  changes  in  the  nonvariant  system  due  to 
addition  and  subtraction  of  heat  and  work,  it  is  in  order  to  treat  in 


One  Component 


1 1 


the  same  way  the  three  monovariant  systems,  liquid  and  vapor,  solid 
and  vapor,  liquid  and  solid.  According  to  the  Phase  Rule  these 
systems  may  exist  over  a  series  of  temperatures  and  a  series  of  pres- 
sures, bounded  only  by  the  appearance  of  new  phases ;  but  if  the 
temperature  is  fixed,  the  pressure  is  also  fixed  and  vice  versa.  This 
is  the  case  experimentally  and  in  Table  II l  are  the  values  at  differ- 
ent temperatures  of  the  corresponding  pressures  for  several  examples 
of  the  monovariant  system,  liquid  and  vapor.*  The  fact  of  equilib- 
rium between  liquid  and  vapor  being  possible  at  different  tempera- 
tures is  familiar  to  everyone,  but  that  the  pressure  is  constant  for 
constant  temperature  can  be  known  only  by  quantitative  measure- 
ments. 

TABUS  II 


Ether 

Alcohol 

Water 

lodbenzene 

Mercury 

0° 

184.9                      12.2 

4-6 

10 

291.8 

23-8 

9-i 

20 

442.4 

44.0 

17.4 

O.OOI 

30 

647.9 

78.1 

3i-5                i-5 

0.003 

40 

921.2 

133-4 

54-9                2.7 

0.006 

5<> 

I276.I 

219.8 

92.0                4.8 

0.013 

60 

I728.I 

350.2 

148.9                 8.2 

0.026 

70 

2273.9 

540-9 

233-3               13-6 

0.050 

80 

2991.4 

811.8 

354.9              21.6 

0.093 

90 

3839.7 

1186.5 

525-5              33-5 

0.165 

loo           4859-° 

1692.3 

760.0              50.4 

0.285 

•  If  the  system,  liquid  and  vapor,  be  subjected  at  constant  tempera- 
ture to  an  external  pressure  always  greater  than  its  own,  there  \vill 
be  an  increase  in  the  denser  or  liquid  phase,  the  vapor  condensing 
until  that  phase  has  disappeared  and  there  is  present  the  divariant 
system,  liquid.  If  the  external  pressure  be  kept  constantly  less  than 
that  of  the  system,  and  the  temperature  not  allowed  to  change,  the 
liquid  will  evaporate  until  there  is  left  only  the  vapor  phase.  Water 
in  an  open  vessel  in  a  large  room  will  evaporate  completely  because 
the  concentration  of  water  vapor  in  the  room  is  less  than  that  in 


1  The  pressures  are  given  in  millimeters  of  mercury. 

*-  For  some  interesting  data  cf.  Barus,  Phil.  Mag.  (5)  29,  141  (1890). 


12  The  Phase  Rule 

equilibrium  with  liquid  water.  Under  a  bell  jar  the  water  evaporates 
until  the  equilibrium  pressure  is  reached  and  then  stops.  If  heat  be 
added  to  the  system  kept  at  constant  pressure,  the  vapor  phase  will 
increase  at  the  expense  of  the  liquid  phase  without  rise  of  tempera- 
ture till  the  latter  has  disappeared,  a  change  which  is  accompanied 
by  absorption  of  heat ;  if  heat  is  subtracted  under  the  same  circum- 
stances, the  system  will  pass  also  without  change  of  temperature,  into 
the  di variant  system  consisting  of  a  liquid  phase  only.  These  two 
changes  can  take  place  at  any  temperature,  but  there  is  a  form  of 
the  first  one,  occurring  experimentally,  which  seems  at  first  sight  to 
be  connected  with  a  definite  temperature.  If  a  liquid  is  heated  in  an 
open  vessel  there  seems  to  be  no  change  beyond  a  rise  of  tempera- 
ture until  the  boiling  point,  so-called,  is  reached,  when  the  liquid 
distills  off,  the  temperature  remaining  constant.  In  the  first  place, 
the  liquid  evaporates  at  all  the  intermediate  temperatures,  though 
this  is  not  noticed,  the  quantity  being  usually  small  under  the  con- 
ditions of  the  experiment.  In  the  second  place,  there  is  air  in  the 
vessel,  so  that  we  are  no  longer  considering  a  system  made  up  of  one 
component.  For  all  that,  it  is  better  to  treat  the  subject  here  rather 
than  later,  since  the  air  is  really  only  a  disturbing  element  and  not  an 
integral  part  of  the  system.  The  temperature  between  the  system 
and  the  heat  reservoir — the  Bunsen  burner,  for  instance — is  so  great 
that  the  liquid  vaporizes  faster  than  it  can  diffuse  out  of  the  vessel.1 
The  system  acts  to  a  certain  extent  as  if  it  were  receiving  heat  while 
at  constant  volume,  the  temperature  rising  and  the  vapor  pressure 
increasing.  The  vapor  phase  in  the  vessel,  being  air  at  atmospheric 
pressure  plus  the  vapor  of  the  liquid,  is  at  a  higher  pressure  than  the 
external,  atmospheric  pressure  and  diffuses  out  against  it.  The 
vapor  phase  becomes  ever  richer  in  the  vapor  of  the  liquid  and  poorer 
in  air  till  all  the  air  has  been  driven  out  and  there  is  a  true  mono- 
variant  system,  liquid  and  vapor.  This  occurs  when  the  vapor  pres- 
sure of  the  liquid  is  equal  to  the  atmospheric  pressure.  If  the  at- 
mospheric pressure  does  not  change,  we  have  a  monovariant  system 
at  constant  pressure  and  the  temperature  will  remain  constant  until 


1  In  the  spheroidal  state,  on  the  other  hand,  the  evaporation  is  so  rapid  that 
water  does  not  rise  to  its  boiling  point.     Cf.  Ramsay  and  Young,  Phil.  Trans. 


O*f  Component  13 

the  liquid  phase  has  disappeared.  If  the  atmospheric  pressure  be 
changed  in  any  way  the  phenomena  of  boiling  will  occur  at  the  tem- 
perature at  which  the  vapor  pressure  is  equal  to  the  modified  atmos- 
pheric pressure.  If  the  external  pressure  be  changed  sufficiently 
the  liquid  may  be  made  to  boil  at  any  temperature  at  which  it  can 
exist  in  equilibrium  with  the  vapor,  from  the  inversion  to  the  critical 
temperature.1  If,  instead  of  allowing  the  liquid  to  distill  off,  the 
flask  is  connected  with  a  reverse  cooler  so  that  the  condensed  liquid 
runs  back,  there  is  only  vapor  of  the  liquid  in  the  flask,  while  outside 
there  is  air.  and  it  may  be  asked  why  the  air  does  not  diffuse  into 
tile  flask.  There  is  a  tendency  for  it  to  do  this  but  the  air  is  carried 
back  by  the  constant  current  of  vapor  streaming  up.  There  will  be 
a  continual  decrease  in  concentration  of  the  air  from  outside  to  the 
point  where  the  vapor  pressure  of  the  liquid  is  equal  to  the  atmos- 
pheric pressure.  If  the  cooling  is  sudden  so  that  the  liquid  con- 
denses all  at  one  place,  there  will  be  an  abrupt  change  from  all  air 
to  all  vapor.  If  the  condensation  takes  place  along  quite  an  inter- 
val, as  usually  happens  with  glass  condensers,  there  will  be  a  grad- 
ual transition,,  the  amount  of  air  decreasing  and  that  of  vapor  in- 
creasing as  one  approaches  the  liquid.  In  other  words,  when  the 
pressure  of  a  saturated  vapor  is  kept  equal  to  the  external  pressure, 
the  vapor  is  impermeable  to  other  vapors  and  gases.  The  determi- 
nation of  the  boiling  point  in  a  flask  with  a  reverse  cooler  is  open  to 
one  objection  due  to  the  influence  of  gravity.  If  the  liquid  is  boil- 
ing regularly  and  the  flame  underneath  be  turned  up,  the  rate  of 
vaporization  win  be  increased  and  the  vapor  will  condense  at  some 
higher  point  in  the  cooler.  This  will  be  accompanied  by  a  slight 
rise  of  temperature  which  if  not  taken  into  account  may  produce 
errors  when  using  the  Beckmann  boiling-point  apparatus.  The  way 
to  avoid  this  is  to  regulate  the  gas  pressure  or  the  cooling  in  the  con 
denser  so  that  the  precipitation  may  take  place  always  at  the  same 
point. 

If  heat  be  added  to  a  system,  liquid  and  vapor,  kept  at  constant 
volume  there  will  be  an  increase  of  vapor  at  the  expense  of  the 
liquid,  that  change  absorbing  heat,  and  the  vapor  pressure  will  in- 

1 A  definition  of  critical  temperature  is  given  o«  the  next  page. 


14  The  Phase  Ride 

crease.  The  system  can  not  be  in  equilibrium  with  the  new  vapor 
pressure  at  the  same  temperature  and  the  temperature  rises  with  ab- 
sorption of  heat.  This  will  continue  in  many  cases  until  the  liquid 
has  disappeared  and  there  is  present  the  di variant  system,  vapor.  If 
the  volume  of  the  liquid  is  large  relatively  to  that  of  the  vapor,  the 
course  of  events  is  somewhat  different.  On  adding  heat  there  will 
be  an  increase  of  temperature  and  pressure  until  a  definite  tem- 
perature and  pressure  has  been  reached  when  the  surface  of 
the  liquid  which  has  been  fairly  clearly  defined  hitherto,  sud- 
denly billows  up  and  disappears,  the  contents  of  the  vessel  be- 
coming homogeneous.  On  cooling  the  reverse  change  takes 
place,  a  tumultuous  commotion  in  the  tube  and  the  formation  anew 
of  two  phases.  Whether  the  contents  of  the  vessel  are  vapor  or 
liquid  is  impossible  to  determine  because  the  two  are  identical.  The 
temperature  and  pressure  at  which  this  phenomenon  takes  place  are 
known  as  the  critical  temperature  and  pressure.  If  the  system  be 
heated  above  this  temperature  and  allowed  to  expand,  it  passes  with- 
out discontinuity  into  what  is  certainly  the  gaseous  state  ;  if  the 
pressure  is  increased  and  the  system  allowed  to  cool  it  passes  also 
without  discontinuity  into  what  is  unmistakably  the  liquid  phase.1 

TABLE  III 


Hydrogen 

-234-5° 

20. 

Nitrogen 

—  146.0 

35-0 

Oxygen 

—  118.8 

50.8 

Carbonic  acid 

31.0 

77.0 

Ether 

194.4 

35-6 

Acetone 

234-4 

60. 

Alcohol 

243.6 

62.8 

Chloroform 

260.0 

54-9 

Benzene 

288.5 

47-9 

Water 

365- 

200.5 

Andrews,  Phil.  Trans.  2,  575  (1869) ;  Sajontchewsky,  Beibl.  3,  741  (1879)  ; 
Nadeshdin,  Ibid.  8,  721  (1884) ;  Cailletet  and  Colardeau,  Comptes  rendus,  112, 
563  (1891);  Altschul,  Zeit.  phys.  Chem.  II,  577  (1893)  ;  Galitzine,  Wied.  Ann. 
50,521  (1893);  van  der  Waals.  "  Die  Kontinuitat  des  gasformigen  und  fliissi- 
gen  zustandes."  Leipzig  (1881)  ;  Cf.  Landolt  and  Bernstein's  Tabellen,  91. 


One  Component  15 

At  the  critical  temperature  the  liquid  and  vapor  have  the  same 
density.  In  Table  III  are  the  critical  temperatures  and  pressures  of 
several  substances.1  The  pressures  are  given  in  atmospheres. 

This  difference  in  behavior  of  a  substance  above  and  below  the 
critical  temperature  renders  it  possible  to  make  a  distinction  between 
a  gas  and  a  vapor,  a  vapor  being  a  gas  below  its  critical  temperature 
and  a  gas  a  vapor  above  that  temperature.  A  vapor  can  be  con- 
densed to  a  liquid  by  increase  of  pressure  without  change  of  temper- 
ature while  a  gas  cannot  be  ;  it  must  be  cooled  as  well.  This  dis- 
tinction has  been  of  great  importance  in  the  attempts  to  liquefy  sub- 
stances hitherto  known  onl}-  in  the  gaseous  state.  It  had  been  found 
that  certain  substances  were  liquefied  by  subjecting  them  to  great 
pressures  and  it  was  assumed  that  with  sufficiently  high  pressure  any 
gas  could  be  liquefied.  All  attempts  in  this  direction  proved  futile 
with  such  gases  as  nitrogen,  oxygen  and  hydrogen.  After  the  ex- 
periments of  Andrews  on  carbonic  acid  the  reason  for  this  failure 
became  clear.  By  cooling  below  the  critical  temperature  and  using 
high  pressures  it  has  been  possible  to  liquefy  all  the  so-called  per- 
manent gases.2 

Starting  from  the  nonvariant  system,  solid,  liquid  and  vapor,  we 
may  cause  the  liquid  phase  to  disappear  leaving  the  monovariant 
system,  solid  and  vapor.  For  this  system  as  for  all  monovariant 
systems  the  statement  holds  true,  that  for  each  temperature  there  is 
a  definite  pressure  under  which  the  system  is  in  equilibrium  ;  but  this 
is  not  very  easy  to  show  from  direct  measurements  owing  to  the 
small  vapor  pressures  of  most  solids.  - 

In  Table  IV  are  the  determinations  of  the  vapor  pressures  of  ice, 
benzene,  acetic  acid  and  camphor  at  different  temperatures.  The 
pressures  are  given  in  millimeters  of  mercury. 


'Heilborn,  Zeit.  phys.  Chem.  7,  601  (1891). 

-  Faraday,  Phil.  Trans.  113, 160(1823);  *45» T  (1845);  Natterer,  Sitzungsber. 
Akad.  Wiss.  Wien,  5,  351  (1850);  6,  557  (1851);  12,  ^99  (.1854);  Cailletet, 
Ann.  chini.  phys.  (5)  15,132  (1878)  ;  Pictet  Ibid.  13,  145  (1878)  ;  Wroblewski 
and  Olszewski,  Wied.  Ann.  20,243  (i883X';  Olszewski,  Phil.  Mag.  (5)  39,  237  ; 

40,  202  (  1895). 


1  6                                          The  Phase  Rule 

TABLE  IV 

Water 

Benzene 

Acetic  Acid 

Camphor 

Temp. 

Pressure 

Temp. 

Pressure 

Temp. 

Pressure   ! 

Temp. 

Pressure 

-10° 

2.03 

0.° 

24.42 

I.85° 

2-35 

20.° 

1.0 

—   8 

2-37 

I. 

26.18 

6.4I 

3-75 

35- 

1.8 

—  6 

2.8l 

2. 

28.08 

9.l6 

4.70 

62.4 

6-4 

—  4 

3-33 

3. 

30-03 

12.  IO 

6.05 

78.4 

9-5 

-  3 

3-62 

4- 

32.32 

13.30 

6-75 

100. 

22.6 

-  2          3.94         5. 

34.65 

I4-30 

7.20 

132. 

78.1 

—   i          4.28         5.3 

35-41 

15.80 

8.85 

154- 

188.8 

o          4.64 

16.41 

9-45 

175- 

354- 

The  most  interesting  experiments  showing  the  qualitative  exist- 
ence of  a  vapor  pressure  are  those  of  Demarcay,1  of  Hallock2  and  of 
Spring.3  Demarcay  placed  the  metals  to  be  examined  in  a  tube  con- 
nected with  a  Sprengel  pump.  Heating  one  end  of  the  tube  and 
cooling  the  other  he  sublimed  cadmium  at  160°,  zinc  at  184°,  anti- 
mony and  bismuth  at  292°,  lead  and  tin  at  360°.  Hallock  found 
that  sulfur  combined  with  copper  and  other  metals  even  when 
separated  from  them  by  a  long  tube  filled  with  plugs  of  cotton  wool 
to  prevent  convention  currents.  Spring  placed  zinc  and  copper  near 
together  but  not  in  contact  and  observed  the  formation  of  brass  at 
temperatures  well  below  the  melting  point  of  the  more  fusible  metal. 

If  work  be  subtracted  from  the  system  by  keeping  the  external 
pressure  always  a  little  less  than  the  equilibrium  pressure  there  will 
be  increase  of  the  vapor,  the  less  dense  phase,  at  the  expense  of  the 
solid  phase  ;  the  solid  will  evaporate  until  there  is  formed  the  di- 
variant  system,  vapor.  The  evaporation  of  ice  in  a  cold,  dry  room 
is  a  well-known  phenomenon.  If  work  be  added  to  the  system 
there  will  be  condensation  of  vapor  until  there  is  only  solid  present. 

If  heat  be  added  to  or  taken  from  the  system  while  it  remains  at 
constant  pressure,  the  solia  will  evaporate  or  the  vapor  condense  as 
the  case  may  be  without  change  of  temperature.  The  change  when 

1  Cotnptes  rendus,  95,  183  (1882). 
'2  Am.  Jour.  Sci.  (3)  37,  402  1889. 
3Zeit.  phys.  Chem.  15,  76  (1894). 


One  Component  17 

heat  is  added  is  in  the  opposite  direction  to  that  taking  place  on  ad- 
dition of  work.  If  heat  be  added  and  the  system  kept  at  constant 
volume  there  wfll  be  increase  of  temperature  and  of  pressure  with 
formation  of  the  vapor  phase  at  the  expense  of  the  solid.  If  there 
is  relatively  little  of  the  solid  phase,  the  transition  wfll  be  to  the  di- 
variant  system,  vapor  ;  otherwise  to  the  nonvariant  system,  solid, 
liquid  and  vapor.  It  was  found  during  the  discussion  of  the  equili- 
brium between  liquid  and  vapor  that,  in  the  presence  of  air,  the  sys- 
tem behaved  to  a  certain  extent  as  if  it  were  kept  at  constant  volume 
until  the  pressure  of  the  vapor  equalled  the  external  pressure.  This 
is  the  case  in  the  equilibrium  between  solid  and  vapor.  If  ammo- 
nium chloride  be  heated  in  a  test  tube,  the  temperature  wfll  rise  un- 
til the  pressure  of  the  ammonium  chloride  vapor  is  equal  to  the 
barometric  pressure  and  then  the  solid  sublimes  without  further 
change  of  temperature.  By  altering  the  external  pressure  the  sub- 
limation temperature  wfll  change  just  as  the  boiling  temperature 
changes.  If  the  sublimation  temperature  be  lower  than  the  inver- 
sion temperature  the  solid  will  sublime ;  if  it  be  higher  the  solid 
wfll  melt.  Ammonium  chloride  melts  when  heated  under  sufficient 
pressure  ;  ice  sublimes  if  the  external  pressure  is  four  millimeters  of 
mercury  or  less.1 

The  monovariant  system ,  solid  and  liquid,  is  characterized  like  the 
others  by  being  fully  determined  when  the  pressure  or  temperature 
is  fixed  or  the  density  of  either  of  the  phases.  Increase  of  external 
pressure  produces  in  some  cases  disappearance  of  the  liquid,  in  others 
of  the  solid  phase.  In  all  instances,  as  predicted  by  the  Theorem  of 
Le  Chatelier,  it  is  the  less  dense  phase  which  disappears,  the  solid 
in  the  case  of  ice  and  water,  the  liquid  with  most  substances.  De- 
creasing the  external  pressure  produces  the  reverse  change.  Adding 
heat  causes  disappearance  of  the  solid  phase  without  change  of  tem- 
perature if  the  system  be  kept  at  constant  pressure.  If  the  system 
be  kept  at  constant  volume  the  addition  of  heat  causes  transition  to 
the  nonvariant  system,  solid,  liquid  and  vapor  with  rise  of  temper- 
ature and  fall  of  pressure  if  the  solid  is  less  dense  than  liquid,  transi- 
tion to  the  divariant  system,  liquid,  with  rise  of  temperature  and 


and  Young.     Phfl.  Trans.  175,  37  (1884). 


1 8 


The  Phase  Rule 


pressure  if  the  solid  is  denser  than  the  liquid.  Taking  heat  from 
the  system  leads  in  the  first  case  to  the  solid  phase,  in  the  second  to 
the  nonvariant  system.  The  temperatures  at  which  solid  and  liquid 
can  be  in  equilibrium  are  lower  than  the  inversion  temperature  if 
the  solid  is  less  dense  than  the  liquid  and  are  higher  if  the  contrary 
is  true.  The  freezing  point  of  a  substance  being  defined  as  the 
temperature  at  which  solid  and  liquid  can  be  in  equilibrium,  it  is 
lowered  by  increased  pressure  if  the  liquid  is  denser  than  the  solid,1 
otherwise  it  is  raised.2  In  all  cases  it  is  found  experimentally  that 
there  is  only  one  temperature  for  each  pressure  at  w?hich  the  solid 
and  liquid  are  in  equilibrium.  In  Table  V  are  some  of  the  experi- 
mental data  on  the  change  of  the  freezing  point  with  the  pressure.3 
The  pressures  are  given  in  atmospheres  ;  under  the  heading  temper- 
ature are  given  the  changes  in  temperature,  the  melting  point  at 
atmospheric  pressure  being  taken  as  the  standard. 

TABLE  V 


Water                        Naphthalene 

Naphthylamine 

Pressure 

Temperature 

Pressure    Temperature 

Pressure 

Temperature 

O.O6 
8.1 
16.8 

4-  0.0066° 
-0.059 
—  o.  129 

8.0           +P.282 
12.0           +0.405 

8.0 

12.  0 

+  0.105 
+  0.  180 

It  is  unknown  whether  there  is  a  temperature  and  pressure  at 
which  solid  and  liquid  become  indistinguishable  as  is  the  case  for 
liquid  and  vapor  ;  but  it  is  probable  that  there  is  such  a  point  and 
that  the  systems,  solid  and  liquid,  liquid  and  vapor,  can  exist  only 
between  two  critical  temperatures. 4 


1 W.  Thomson,  Phil.  Mag.  (3)  37,  123  (1850)  ;  Dewar,  Proc.  Roy.  Soc.  30, 
533  (1880). 

2Battelli,  AttidelR.  1st.  Ven.  (3)3  ( 1886);  Amagat,  Comptes  rendus,  105, 
165  (1887). 

3Cf.  Ostwald,  Lehrbuch  I,  1013-1015. 

4  Cf.  Voigt,  Kompendium  der  Physik  I,  583.  If  it  were  not  for  the  capillary 
phenomena  this  would  occur  when  solid  and  liquid  had  the  same  density. 
Experimentally,  this  is  not  the  case.  Cf.  Damien,  Comptes  rendus,  112,  785 
(1891). 


One  Component  19 

There  remain  only  the  divariant  systems,  solid,  liquid,  and  vapor 
to  consider.  Vapors  and  gases  fill  the  vessel  in  which  they  are  con- 
tained uniformly,  barring  the  influence  of  gravity,  and  exert  a  pres- 
sure on  the  walls.  For  any  temperature  there  is  possible  any  pres- 
sure so  long  as  no  new  phase  appears,  and  for  any  pressure,  any 
temperature.  Increase  of  external  pressure  produces  decrease  of 
volume  and  vice  versa.  At  constant  pressure  the  addition  of  heat 
produces  an  increase  of  volume  and  a  rise  of  temperature,  both 
changes  involving  an  absorption  of  heat.  At  constant  volume 
addition  of  heat  causes  rise  of  temperature  and  of  pressure.  It 
is  to  be  noticed  in  all  cases  where  there  is  one  phase  and 
one  component,  that  keeping  the  mass  and  the  volume  con- 
stant is  the  same  as  fixing  the  concentration  and  there  is  then 
only  one  degree  of  freedom  left,  one  independent  variable. 
For  each  temperature  there  will  be  only  one  pressure  possible 
for  a  given  concentration.  It  may  be  noted,  in  passing,  as  an  ex- 
perimental fact  that  the  same  concentrations  of  two  gases,  expressed 
in  grams  per  liter  for  instance,  do  not  give  the  same  pressure  at  the 
same  temperature.  The  explanation  of  this  together  with  the 
mathematical  statement  of  it  in  the  form  of  the  Gas  Theorems  be- 
longs under  the  head  of  Quantitative  Kquilibrium  and  will  not  be 
taken  up  in  this  book.  We  have  found  heretofore  that  addition  of 
heat  to  the  system  kept  at  constant  pressure  has  produced  no  change 
in  temperature  and  the  behavior  of  a  vapor  seems  to  be  an  exception. 
This  is  only  apparent,  however.  In  the  previous  cases  there  have 
been  always  two  or  more  phases,  and  there  has  been  a  transference 
of  matter  from  one  phase  to  another  without  change  of  concentra- 
tion in  any  of  the  phases.  This  can  not  happen  where  there  is  only 
one  phase  and  the  volume  changes,  which  always  occur  when  heat  is 
added  to  a  system  kept  at  constant  pressure,  produce  a  change  in  the 
concentrations.  If  the  pressure  is  fixed,  there  is  only  one  independ- 
ent variable  ;  when  the  concentrations  remain  constant  the  tempera- 
ture remains  constan^  and  when  the  concentrations  change  the  tem- 
perature changes.  Addition  of  heat  to  a  system  kept  at  constant 
pressure  produces  no  change  in  temperature  if  the  concentrations  of 
the  phases  remain  unchanged,  a  rise  of  temperature  if  the  concen- 
trations change.  This  is  not  yet  entirely  satisfactory  because  it 


2o  The  Phase  Rule 

leaves  undecided  the  question  under  what  circumstances  the  concen- 
trations do  or  do  not  change.  In  a  nonvariant  system  there  is  no 
change  of  pressure,  temperature  or  concentrations.  In  a  mono- 
variant  system  the  temperature  and  the  concentrations  can  not 
change  when  the  pressure  remains  constant.  No  change  of  temper- 
ature is  produced  when  heat  is  added  to  a  nonvariant  or  a  mono- 
variant  system  kept  at  constant  pressure  ;  in  all  other  cases  there  is 
a  rise  of  temperature.  It  may  be  interesting  to  consider  what 
changes  in  pressure  and  volume  will  take  place  on  adding  heat,  the 
temperature  remaining  constant.  Since  all  systems  expand  on  addi- 
tion of  heat  when  there  is  no  change  of  phase,  there  will  be  always 
an  increase  of  volume  which  carries  with  it,  by  the  Theorem  of  Le 
Chatelier,  a  decrease  in  pressure. 

Liquids  differ  from  gases  and  vapors  in  that  they  do  not  neces- 
sarily fill  the  whole  of  the  vessel  which  contains  them,  and  in  that 
they  have  a  form  of  their  own.  Since  liquids  fill  the  lower  part  of 
the  containing  vessel  completely  and  are  bounded  on  the  upper  side 
by  a  horizontal,  nearly  plane  surface,  it  might  be  thought  that  they 
had  no  definite  shape  of  their  own.  This  behavior  is  due  to  the  in- 
fluence of  gravity,  and  when  this  is  eliminated,  by  suspending  the 
liquid  in  a  fluid  medium  of  approximately  the  same  density,  we 
perceive  that  the  spherical  shape  is  the  true  form,  characteristic  of 
all  liquids.  For  each  temperature  there  can  be  a  series  of  pressures 
at  which  the  liquid  can  exist  and  for  each  pressure  a  series  of  tem- 
peratures, limited  in  both  cases  only  by  the  appearance  of  a  new 
phase.  While  the  change  of  volume  with  change  of  pressure  is 
fairly  large  with  gases  and  vapors '  it  is  very  small  with  liquids,  so 
small  in  fact  that  it  can  be  shown  only  by  careful  quantitative  meas- 
urements.2 

The  solid  phase  is  characterized  by  rigidity  and  elasticity  ;  two 
mutually  exclusive  properties.  By  its  rigidity  it  resists  deformation 
and  is  therefore  not  dependent  on  the  vessel  in  which  it  is  contained 


'Cf.  v.  Lang,  Theor.  Physik.  650;  Ostwald,  Lehrbuch  I,  139-159. 

2  Oersted,  Pogg.  Ann.  9,  603  (1827)  ;  Pagliani  and  Vicentini,  Beibl.  8,  794 
(1884)  ;  Rontgen  and  Schneider,  Wied.  Ann.  29,  165  (1886)  ;  Schumann,  Ibid 
31,  14(1887);  Boguski,  Zeit.  phys.  Cheni.  2,  126(1888);  Amagat,  Jour,  de 
Phys.  (2)  8,  197  (1889). 


One  Componfnt  21 


for  its  shape.  By  its  elasticity  it  returns  to  its  original  shape  after 
having  been  subjected  to  a  deforming  stress.  These  properties  are 
relative  only  and  in  some  cases  almost  imperceptible.  A  steel  spring 
goes  back  to  its  original  form  even  after  having  been  very  much 
compressed  ;  a  piece  of  putty  remains  in  the  new  shape.  The  rig- 
idity varies  very  much  also.  All  solids  flow  a  little  if  left  in  a  state 
of  strain.  The  behavior  of  sealing-wax  is  well  known.  Car-axles 
become  changed  in  structure  through  the  continual  jolting.  Spring  ' 
has  shown  that,  if  two  clean  metal  surfaces  be  brought  into  intimate 
contact,  they  unite  by  diffusion,  and,  in  many  cases,  the  bars  thus 
formed  can  be  placed  in  a  lathe  with  one  end  free  and  have  shavings 
turned  from  them  without  breaking.  When  the  sticks  were  broken 
by  twisting,  the  fracture  did  not  come  at  the  junction  but  usually 
across  it.  While  a  copper  wire  bends  without  breaking,  showing  a 
power  of  internal  readjustment,  a  stick  of  bismuth  is  so  brittle  that 
it  fractures  under  a  very  slight  strain.  Whether  the  crystalline 
structure  or  tendency  to  assume  definite  shapes  bounded  by  plane 
surfaces  is  a  characteristic  of  solids  is  a  doubtful  question.  Solids 
certainly  occur  in  what  is  known  as  the  amorphous  state  in  which 
no  signs  of  a  crystalline  structure  can  be  detected  by  any  means  at 
our  disposal.  Nernst  ~  has  made  the  suggestion  that  the  solid  is 
really  present  in  very  minute  crystals  ;  but  he  offers  little  evidence 
in  behalf  of  this  view,  and  it  is  not  generally  accepted, 

As  predicted  by  the  Phase  Rule,  the  solid  phase  can  exist  at  a 
series  of  temperatures  and  for  each  temperature  at  a  series  of  pres- 
sures limited  only  by  the  appearance  of  new  phases.  Increase  of 
external  pressure  causes  diminution  of  volume,  and  may  cause  ap- 
pearance of  the  liquid  phase  if  the  solid  is  less  dense  «than  the 
liquid.  Decrease  of  external  pressure  is  accompanied  by  expansion 
and  eventually  by  formation  of  one  of  the  uionovariant  systems, 
solid  and  liquid  or  solid  and  vapor,  as  the  case  may  be.  Addition  of 
heat  to  a  solid  kept  under  constant  pressure  produces  increase  of 
volume  and  temperature  with  eventual  formation  of  solid  and  liquid 
or  solid  and  vapor,  depending  on  the  nature  of  the  substance  and 
the  initial  pressure  and  temperature,  If  the  system  is  kept  at  con- 


1  Zeit.  phrs.  Chem.  15,  70  ( 1894). 
-Theor.  Chem.  65. 


22  The  Phase  Rule 

stant  volume  addition  of  heat  causes  increase  of  temperature  and 
pressure  and  transition  to  the  monovariant  system,  solid  and  liquid. 

A  system  is  said  to  be  in  stable  equilibrium  when  the  addition 
of  any  modification  of  any  of  the  components  produces  a  change  in 
the  system  proportional  to  the  quantity  of  substance  added.1  The 
equilibria  which  we  have  considered  so  far  have  all  been  of  this 
type.  If  the  solid  phase  be  added,  for  instance,  to  the  monovariant 
system,  liquid  and  vapor,  the  solid  will  melt  except  at  the  inversion 
temperature,  when  it  is  in  equilibrium.  The  changes  in  either 
case  will  be  proportional  to  the  quantity  of  solid  added. 

It  is  possible  to  have  a  system  in  equilibrium  with  respect  to  the 
phases  then  co-existing  which  shall  not  be  in  equilibrium  when 
brought  in  contact  with  some  other  modification  of  the  components. 
Such  a  system  is  said  to  be  in  labile  equilibrium  because  the  equili- 
brium though  stable  as  regards  the  phases  already  present  is  instable 
with  respect  to  some  other  phase.2  The  most  familiar  example  of 
this  is  the  supersaturated  solution  of  sodium  sulfate  which,  if  left  to 
itself,  will  remain  unchanged  for  months,  perhaps  years.  If  a  crys- 
tal of  the  hydrated  salt  be  thrown  in,  there  is  a  sudden  crystalliza- 
tion and  the  quantity  of  the  new  phase  formed  bears  no  relation  to 
the  amount  of  salt  added  to  start  the  reaction.  These  labile  equili- 
bria occur  in  all  systems,  and  we  will  take  up  first  the  supercooled 
vapors.  If  a  vapor  be  cooled  at  constant  volume,  there  will  be 
reached  a  temperature  and  pressure  at  which  there  should  be  forma- 
tion of  liquid,  and  this  usually  takes  place.  It  is  possible,  however, 
by  careful  cooling,  especially  if  there  be  no  dust  present,  to  pass  this 
point  without  the  liquid  phase  being  formed.3  This  equilibrium  is 
now  labile,  for  the  addition  of  the  smallest  quantity  of  liquid  pro- 
duces a  sudden  condensation  which  ceases  only  when  the  vapor 
pressure  characteristic  of  the  monovariant  system  at  that  tempera- 
ture has  been  reached.  A  second  form  of  labile  equilibrium,  that  of 


1  Gibba,  Trans.  Conn.  Acad.  3, 455  (1878);  Meyerhoffer,  Die  Phasenregel,  10. 

2  It  is  a  step  back  ward  to  class   a  supersaturated   solution  and  a  mixture  of 
hydrogen  and  oxygen  under  the  same  head  as  Duhem  (Me"canique  chimique, 
158)  has  done.     Addition  of  water  as  vapor  or   liquid  to  the  hydrogen  and  oxy- 
gen mixture  produces  no  change      It  is  not  a  case  of  labile  equilibrium. 

SR.  v.  Helmholtz,  Wied.  Ann.  27,  521  (1886). 


One  Component  23 

a  superheated  liquid,  is  even  harder  to  observe  experimentally. 
Donny  *  has  shown  that  it  is  possible  by  unequal  heating  to  raise  the 
temperature  of  liquid  water  to  138°  without  its  boiling.  This  can 
be  done  only  by  keeping  the  surface  of  the  water  cool  so  that  the 
superheated  liquid  is  not  in  contact  with  the  vapor.  Dufour*  at- 
tained the  same  result  by  suspending  drops  of  water  in  a  mixture  of 
linseed  oil  and  oil  of  cloves  having  the  same  density  as  water.  A 
temperature  of  175°  was  reached  in  this  way.  The  third  volume  of 
van  der  Waal's*  is  a  mathematical  fiction  and  there  seems  to  be  no 
good  experimental  ground  for  assuming  that  the  vapor  pressures  of 
superheated  liquids  and  supercooled  vapors  are  parts  of  the  same 
curve.  It  is  possible  to  obtain  another  instance  of  labile  equilibrium 
by  cooling  a  liquid  in  the  absence  of  dust  below  the  inversion  tem- 
perature. With  most  liquids  it  is  possible  to  supercool  the  liquid 
a  degree  or  so  and  with  some  the  supercooling  can  be  carried  much 
farther.*  The  reverse  case  of  a  solid  heated  above  its  melting  point 
has  never  been  realized.5  The  third  monovariant  system,  solid  and 
vapor  has  not  been  studied  with  the  same  care  as  the  other  two,  and 
I  am  not  able  to  cite  any  quantitative  example  of  a  supercooled 
vapor  though  it  iran  hardly  be  an  uncommon  phenomenon.'  The 
' '  hanging ' '  of  mercury  in  barometer  tubes 7  is  an  instance  of  a 
vapor  phase  not  being  formed  on  decrease  of  pressure,  though  this 
is  somewhat  complicated  by  surface  tension  phenomena.  In  all  the 
cases  of  labile  equilibrium  the  presence  of  the  smallest  quantity  of 
the  phase  in  respect  to  which  the  system  is  instable  brings  about  a 
change  bearing  no  relation  in  its  extent  to  the  quantity  of  that  phase. 


lPogg.  Ann.  67,  562  (1846) ;  Cf.    Gernez,  Comptes  rendus,  86,  472  ;  87, 
1549  (1876). 

'Ann.  Chim.  Phys.  (.3)  68,  370  (1863). 

*  Ostwald,  Lehrbuch  I,  299. 

*Schrotter,  Sitzungsber.  Akad.    Wiss.    Wien,  IO,  527  (1853);    Schroder, 
Liebig's  Annaleu  109,  45  (1859). 

5  In  the  case  cited  by  Ostwald  ( Lehrbuch  I,  994),  there  are  two  components. 

•  Cf.  Lehmann,  Molekularphysik,  II,  581. 
7  Hoser,  Pogg.  Ann.  160,  138  ( 1877). 


CHAPTER  III 


WATER,    SULFUR   AXD   PHOSPHORUS 

The  general  results,  which  have  been  enumerated  in  regard  to 
equilibrium,  will  be  grasped  more  easily  if  they  are  represented 
graphically.  In  Fig.  i  are  shown  the  limiting  values  of  pressure 
and  temperature  for  water  in  its  different  modifications.  The  ordi- 
nates  are  pressures  and  the  abscissae  temperatures  ;  the  drawing  is 
not  to  scale. 

1 
P 


FIG.  i. 

The  curve  OA  is  the  vaporization  *  curve,  showing  the  pressures 
and  temperatures  at  which  liquid  and  vapor  can  coexist,  and  it  can 
be  seen  from  the  diagram  that  for  each  temperature  there  can  be 


Bakhnis  Ronzeboom,  Recueil.  Trav.  Pays-Bas  6,  280  (1887). 


One  Component  : 

only  one  pressure  and  for  each  pressure  only  one  temperature  at 
which  these  two  phases  can  be  in  equilibrium.  The  curve  termi- 
nates at  O  because  the  solid  phase  appears  ;  at  the  other  end  rt  is 
limited  by  the  critical  temperature,  of  365°,  and  the  critical  presswrc 
of  200  atmospheres,  the  difference  between  the  two  phases  disap- 
pearing.1 The  curve  OB  is  the  sublimation  curve,  the  sofid  and 
vapor  phase  being  in  equilibrium.  The  curve  OC  represents  the 
equilibrium  between  solid  and  liquid.  In  this  case  it  slants  to  the 
left  because  ice  is  less  dense  than  liquid  water. 

In  the  corresponding  diagram  for  naphthalene,  it  would  slant 
to  the  right  because  the  solid  is  more  dense  than  the  liquid  and  the 
freezing  point  is  therefore  raised  by  pressure.  We  do  not  know 
whether  there  is  an  upper  limit  beyond  which  this  curve  can  not  ex- 
tend. The  curve  OD  is  the  continuation  of  AO  and  represents  the 
labile  equilibrium  between  water  and  vapor.  It  will  be  seen  from 
the  diagram  that  this  curve  lies  above  the  curve  for  ice  and  vapor  ~ 
and  is  therefore  instabie  with  respect  to  itr  as  vapor  will  distill  from 
the  water  and  condense  on  the  ice  until  the  former  disappears.  This 
is  the  easiest  way  of  making  the  facts  intelligible  though  it  is  not 
really  accurate  ;  because  the  change  is  not  one  of  distillation  from  a 
place  of  high  pressure  to  one  of  low  pressure,  but  a  direct  change  of 
liquid  into  solid,  as  appears  from  the  velocity  of  the  reaction.  In 
the  present  state  of  our  knowledge  we  can  express  instability  only 
in  pressure  and  concentration  differences  although  that  is  obviously 
a  very  incomplete  statement  of  things  as  they  are.  The  three  curves 
AO.  BO  and  CO  meet  at  the  point  O,  the  inversion  point,  if  we  neg- 
lect the  effects  due  to  surface  tension.  In  an  actual  system  it  is 
quite  possible  that  this  is  not  the  case1  though  this  has  never  been 
shown  experimentally.  If  the  three  curves  do  not  meet  at  a  point 
the  intersection  of  AO  and  CO  will  give  the  inversion  temperature 
and  pressure.  Since  the  nonvariant  system  can  exist  at  the  point  O 
and  the  monovariant  systems  each  along  one  of  the  curves/  it  fol- 


1  Cailletet  and  Colardeau.  Comptes  rendus,  na,  1170  ( i.Sgi  i. 

*  C£  Ramsay  and  Young,  PfriL  Trans.  175,  II,  461  (1884)  ;  Fercfce,  Wied. 
AML  44,  2-7 

3  Wald,  ZeiL  pfays.  Cn«n.  7,  5*4  ( i»ji>- 

*  Usually  known  as  boundary  carves. 


26  The  Phase  Rule 

lows  that  the  divariant  systems  are  in  stable  equilibrium  in  the  fields 
bounded  by  the  curves.  Vapor  can  exist  at  any  temperature  and 
pressure  in  the  field  AOB,  liquid  in  the  field  AOC,  and  solid  in  the 
field  COB.  The  diagram  summarizes  also  what  we  have  learned 
about  the  effect  of  addition  or  subtraction  of  heat  and  work.  If  we 
start  at  some  point  M  in  the  field  AOB  where  the  pressure  is  greater 
than  the  inversion  pressure  and  subtract  heat,  keeping  the  pressure 
constant,  the  temperature  will  fall  until  the  point  Mj  is  reached, 
when  the  liquid  phase  appears  and  the  temperature  remains  constant 
until  the  whole  of  the  vapor  phase  has  disappeared.  The  tempera- 
ture will  fall  again  until  at  the  point  M.,  the  solid  phase  appears  and 
the  temperature  begins  to  decrease  once  more  only  when  the  liquid 
phase  has  disappeared.  If  we  start  from  the  point  H  and  subtract 
heat  while  keeping  the  system  at  constant  volume,  the  pressure  and 
temperature  will  change  in  a  way  depending  on  the  equation  of  con- 
dition for  the  vapor.  This  is  indicated,  without  any  attempt  at 
accuracy,  by  the  line  HH,.  At  Ht  the  vapor  is  in  equilibrium  with 
the  liquid  ;  but,  if  the  cooling  is  done  carefully,  it  is  possible  to  pre- 
vent condensation  and  to  realize  a  small  portion  of  the  dotted  curve 
HjK  when  the  vapor  is  in  labile  equilibrium.  Ordinarily  condensa- 
tion takes  place  at  Hj  and  there  is  formed  the  monovariant  system, 
liquid  and  vapor.  The  boundary  curves  are  curves  for  equilibrium 
at  constant  volume  so  long  as  there  is  a  monovariant  system  present. 
On  further  subtraction  of  heat,  the  pressure  and  temperature  fall, 
the  corresponding  values  always  lying  on  the  curve  H,O.  At  O  the 
solid  phase  appears  and  both  temperature  and  pressure  remain  con- 
stant until  one  of  the  phases  has  disappeared.  Usually  this  will  be 
the  liquid  phase  and  subsequent  pressures  and  temperatures  are  rep- 
resented by  the  line  OB.  In  the  case  of  water  the  vapor  phase  may 
disappear  if  its  volume  is  small  in  comparison  with  that  of  the  liquid, 
whereupon  further  subtraction  of  heat  will  cause  the  system  to  pass 
along  the  curve  OC,  the  temperature  falling  and  the  pressure  in- 
creasing. If  the  solid  and  vapor  are  cooled  with  due  precautions, 
the  solid  phase  may  not  appear  at  O  and  the  pressures  and  tempera- 
tures of  supercooled  water  are  observed,  represented  by  the  dotted 
line  OD.  If  we  start  from  a  third  point  N  in  the  field  AOB  and 
increase  the  external  pressure,  keeping  the  temperature  constant, 


One  Component  27 

the  diagram  tells  us  that  the  pressure  of  the  system  may  increase  in- 
definitely, remaining  constant,  however,  while  the  vapor  condenses 
to  ice  at  N\  and  while  the  ice  melts  to  water  at  X,.  From  this  brief 
review  it  is  clear  that  the  diagram  gives  a  concise,  intelligible  sum- 
mary of  all  the  facts  and,  in  treating  other  cases,  it  will  not  be  neces- 
sary to  go  over  the  ground  twice  as  has  been  done  this  time  ;  it  will 
be  sufficient  to  start  from  a  diagram  and  study  that.  It  is  to  be  no- 
ticed that  the  treatment  has  been  entirely  general  and  the  results  are 
independent  of  the  equations  of  condition  describing  or  failing  to 
describe  the  three  phases.  It  is  immaterial  whether  water  vapor 
dissociates  into  hydrogen  or  oxygen  at  the.  temperature  of  the  ex- 
periment or  not,  provided  the  composition  of  each  phase  may  always 
be  represented  empirically  by  the  formula,  HSO.  If  the  system  is 
in  a  state  of  reversible  equilibrium,  it  is  indifferent  whether  there  is 
dissociation,  association  or  neither  in  the  vapor  phase  ;  if  the  system 
is  not  in  equilibrium  it  can  not,  of  course,  be  represented  in  a  dia- 
gram which  gives  equilibrium  pressures  and  temperatures  only.1  It 
is  not  permissible  to  add  an  excess  of  either  hydrogen  or  oxygen  be- 
cause the  system  would  then  contain  two  components  instead  of  one. 
One  word  on  another  point  may  not  be  superfluous.  In  the  previous 
discussion  the  addition  of  work  and  the  addition  of  heat  have  been 
treated  as  if  they  were  two  independent  processes.  This  is  not  true. 
When  work  is  added  to  of,  more  properly,  done  upon  the  system 
there  is  a  heat  effect  produced  in  all  except  adiabatic  changes.  In 
like  manner  when  heat  is  added  to  the  system,  there  is  always  addi- 
tion or  subtraction  of  work  except  in  the  one  case  when  the  volume 
is  kept  constant.  It  has  seemed  advisable  to  concentrate  the  atten- 
tion first  on  the  one  effect  and  then  on  the  other,  ignoring  the  simul- 
taneous manifestations  of  energy  in  other  forms.  These  unconsidered 
energy  changes  are  referred  to  in  the  pro  visions  that  the  temperature 
be  kept  constant  when  the  addition  or  subtraction  of  work  is  under 
consideration  and  that  the  pressure  be  kept  constant  when  the  effects 
produced  by  the  addition  or  subtraction  of  heat  are  the  interesting 
changes. 

1  Nernst  (Theor.  Chem.  487)  does  not  seem  to  be  very  clear  on  this  point. 
He  speaks  of  the  mixture  of  hydrogen  and  oxygen  as  being  in  labile  equili- 
brium, and  in  the  same  paragraph  (with  a  reference  to  p.  532 )  of  its  not  being 
in  equilibrium  at  all. 


The  Phase  Rule 


Water  is  a  compound  which  can  exist  in  only  three  forms,  solid, 
liquid  and  vapor.1  While  there  are  no  substances  known  which 
form  two  liquid  phases,  there  are  quite  a  number  which  can  occur  in 
two  or  more  solid  modifications.  This  brings  in  other  inversion 
temperatures,  and  we  will  take  up  first  the  case  of  sulfur  which 
exists  certainly  hi  the  rhombic  and  mouoclinic  forms  and  possibly 
in  other  modifications.  In  Fig.  2  is  a  graphical  representation  of 
our  knowledge  in  respect  to  the  coexisting  phases  of  sulfur ;  the 
abscissae  denote  temperatures  and  the  ordinates  pressure  as  before: 
The  diagram  is  not  to  scale. 

D 


FIG.  2. 

At  ordinary  temperatures  the  rhombic  crystals  are  the  more 
stable  and  the  curve  O,B  is  the  sublimation  curve  for  rhombic  sulfur. 
O,A  is  the  curve  for  the  monovariant  system,  liquid  and  vapor, 
while  C^C  represents  the  equilibrium  between  rhombic  and  liquid 
sulfur.  In  this  there  is  nothing  different  from  the  behavior  of  water 


1  Cf.  however,  Prendel,  Zeit.  Kryst.  23,  76  (1894). 


One  Component  29 

except  that  the  line  O,C  slants  off  to  the  right.  The  inversion  tem- 
perature at  which  rhombic  and  liquid  sulfur  can  exist  in  presence  of 
vapor  is  1 14.5°.  This  is  not  the  only  inversion  temperature.  The 
curve  O,B,  represents  the  equilibrium  between  monoclinic  sulfur  and 
vapor,  O,A  that  between  liquid  sulfur  and  vapor,  and  O,C  that  be- 
tween monoclinic  and  liquid  sulfur.  The  temperature  of  O,  is  120°. 
The  curves  O,B  and  O,Bj  intersect  at  O,  giving  a  new  triple  point 
with  the  three  phases,  rhombic  and  monoclinic  sulfur  and  vapor. 
From  this  point  starts  the  curve  OC  showing  the  conditions  of  equil- 
ibrium in  the  mono  variant  system,  rhombic  and  monoclinic  sulfur. 
The  temperature  at  which  this  third  nonvariant  system  exists  is 
95.4°.  The  pressure  has  not  been  determined.  Not  all  of  these 
curves  are  stable.  The  curves  BQ,  for  rhombic  sulfur  and  vapor 
ceases  to  be  stable  at  O,  and  the  remainder  of  the  curve  OO,  repre- 
sents a  labile  equilibrium.  The  pressures  are  higher  than  those  for 
the  vapor  in  equilibrium  with  monoclinic  sulfur,  shown  by  OOJf  and 
addition  of  monoclinic  sulfur  causes  a  complete  conversion  of  the 
rhombic  sulfur  into  the  more  stable  form.  If  monoclinic  sulfur  is 
not  added,  the  change  does  not  take  place  readily  and  the  curve  can 
be  followed  to  the  melting  point  of  rhombic  sulfur  at  H4-50.1  The 
liquid  sulfur  has  a  higher  vapor  pressure  than  the  inonoclinic  sulfur 
at  that  temperature  and  the  curve  O,A  is  therefore  one  of  labile 
equilibrium  as  far  as  Or  The  curve  O,C  for  rhombic  and  liquid  sul- 
fur is  instable  in  respect  to  the  monocUnic  form  and  has  not  been 
studied.  Monoclinic  sulfur  can  exist  in  stable  equilibrium  with  the 
vapor  of  sulfur  from  the  melting  point  at  O^,  120°,  to  the  other 
inversion  point  at  O,  95.4°.  Below  this  temperature,  rhombic  sulfur 
has  the  lesser  vapor  pressure  and  is  the  more  stable  form  ;  above  it, 
this  is  reversed,  while  at  the  temperature  and  pressure  represented 
by  O,  and  at  this  temperature  and  pressure  only  can  rhombic  and 
monoclinic  sulfur  exist  in  stable  equilibrium  with  each  other  and 
with  sulfur  vapor.  The  work  on  this  point  has  been  done  by 
Reicher*  under  van  't  HofFs  direction.  If  there  is  no  vapor  of 
sulfur  present  we  have  a  mono  variant  system  instead  of  a  non- 


•  Brodie,  Phil.  Mag.  (4)  7,  439  (1894)- 

:  Recueil  Trav.  Pays-Bas,  a,  246  (1883)  ;  ZeiL  KijsL  8,  593  ( i8&|). 


30  The  Phase  Rule 

variant  one  and  there  can  therefore  be  equilibrium  between  mono- 
clinic  and  rhombic  sulfur  at  other  temperatures  provided  the  system 
is  under  proper  pressure.  This  is  shown  graphically  by  the  curve 
OC,1  a  few  points  of  which  have  baen  determined  by  Reicher. 
Since  the  rhombic  sulfur  is  the  denser  form,  increase  of  pressure 
must  raise  the  temperature  at  which  it  can  be  in  equilibrium  with 
monoclinic  sulfur.  Reicher  found  that  for  a  pressure  of  four  at- 
mospheres the  corresponding  temperature  was  95.6°,  and  for  a  pres- 
sure of  a  little  less  than  sixteen  atmospheres  96. 2°,  a  change  of  0.6° 
for  twelve  atmospheres.'2  Although  these  results  agree  both  qualita- 
tively and  quantitatively  with  the  calculated  change  of  temperature 
with  the  pressure,3  their  value  is  somewhat  doubtful  because  the 
system  actually  observed  contained  more  than  one  component.  It 
is  by  no  means  certain  that  the  liquids  used  in  the  dilatometer  had 
no  effect  on  the  equilibrium.4 

The  curves  OC,  OjC,  O2C  meet  at  an  unknown  point  provided 
that  no  other  modification  of  sulfur  appears.  The  position  of  this 
point  has  been  calculated  from  the  pitch  of  the  curves  OC  and  O2C 
by  Roozeboom.0  Assuming  that  there  is  no  change  in  the  specific 
heats  of  the  two  solid  modifications  of  sulfur  he  finds  that  the  in- 
version temperature  should  be  about  135°  and  the  pressure  about 
400  atmospheres.  At  the  point  C  there  would  coexist,  rhombic, 
monoclinic  and  liquid  sulfur,  and  at  higher  pressures  the  monoclinic 
sulfur  would  disappear  and  the  stable  monovariant  system  possible 
along  CD  would  be  rhombic  and  liquid  sulfur.  This  has  not  been 
realized  experimentally.  As  in  all  diagrams  of  this  kind  the  di- 
variant  systems  exist  in  the  fields,  we  have  sulfur  vapor  bounded  by 
AOjB,  liquid  sulfur  by  AOjD,  rhombic  sulfur  by  DOjB  and  mono- 
clinic  sulfur  by  O  CO2O.  It  will  be  noticed  that  the  monoclinic  modi- 
fication is  the  only  one  existing  in  a  closed  field  and  that  the  over- 
lapping portions  of  the  other  fields  represent  states  of  labile 
equilibrium,  instable  with  respect  to  monoclinic  sulfur.  The  labile" 

1  This  curve  has  been  followed  over  a  range  of  100°  for  the  two  modifica- 
tions of  silver  iodide.    Le  Chatelier  and  Mallard,  Comptes  reudus,  99,  157  (1884). 
2Recueil  Trav.  Pays-Bas,  2,  262,  269  (1884). 
3  Ibid.  266. 

4Cf.  Bancroft,  Jour.  Phys.  Chem.  I,  No.  3  (1896). 
5Recueil  Trav.  Pays-Bas.  6,  314  (1887). 


One  Component  31 

states  of  equilibrium  in  the  case  of  sulfur  show  the  same  character- 
istics as  the  corresponding  states  in  the  case  of  water.  Addition  of 
a  slight  quantity  of  the  phase  with  respect  to  which  the  system  is 
instable  produces  a  rapid  transformation  ;  but  with  sulfur  there  is  a 
case  not  analogous  to  any  occurring  with  water.  When  the  change 
is  from  one  solid  phase  to  the  other  it  does  not  take  place  instantane- 
ously. Reicher1  found  that  the  reaction  velocity  for  the  change  of 
monoclinic  into  rhombic  sulfur  increased  as  the  temperature  fell  be- 
low 95°,  reaching  a  maximum  at  about  35°  and  then  decreasing. 
Ruys2  took  advantage  of  a  winter  spent  upon  the  sea  of  Kara  to 
observe  the  reaction  velocity  at  a  temperature  of  —35°  and  found 
that  under  those  circumstances  the  time  necessary  for  the  change  of 
a  given  weight  of  monoclinic  into  rhombic  sulfur  was  about  five 
hundred  times  as  long  as  at  ordinary  temperatures.  The  reaction 
velocity  is  a  function  of  the  pressure  difference  and  the  absolute 
temperature.  Below  the  inversion  point  these  work  against  each 
other  and  the  point  of  maximum  velocity  is  the  temperature  at  which 
the  latter  effect  just  overbalances  the  former.  Above  the  inversion 
point  the  two  causes  work  together  and  there  is  no  decreasing 
velocity  after  exceeding  a  given  temperature. 

Since  the  monoclinic  is  the  less  stable  form  at  low  temperatures, 
it  follows  from  the  Theorem  of  Le  Chatelier  that  the  change  to  the 
rhombic  modification  must  be  accompanied  by  an  evolution  of  heat, 
and  this  is  the  case,  experimentally.8  It  is  impossible  to  take  into 
account  the  additions  to  the  diagram  due  to  the  existence  of  other 
modifications  of  sulfur  because  only  the  rhombic  and  monoclinic 
forms  have  been  studied  with  any  approach  to  thoroughness.4 

It  is  not  necessary  that  each  allotropic  form  must  be  stable  at 
some  temperature  and  pressure,  and  a  very  striking  instance  of  this 
is  found  in  the  case  of  phosphorus.5  Fig.  3  is  the  pressure-tempera- 
ture diagram  for  this  substance.  The  drawing  is  not  to  scale. 


'Recueil  Trav.  Pays-Bas,  2,  251  (1883). 

*  Ibid  3,  i  (1884). 

*  Mitscherlich,  Pogg.  Ann.  88,  328  (1853). 

4  Cf.  Damtner,  Handbuch  I,  597-605  ;  Meslans,  Etats  allotropiques  des 
corps  simple,  34  ;  D.  Berthelot,  Allotropie  des  corps  simple,  18. 

&Bakhuis  Roozeboom,  Recueil  Trav.  Pays-Bas,  6,  272(1887);  Riecke's 
treatment  is  very  inaccurate.  Zeit.  phys.  Chem.  6,  411  (1890)  ;  7,  115  (1891). 


The  Phase  Rule 


FIG.  3. 

AB  is  the  boundary  curve  for  red  phosphorus  and  vapor,  ED 
for  yellow  phosphorus  and  vapor  and  DC  for  liquid  phosphorus  and 
vapor.  The  curve  CDE  lies  above  the  curve  BA  and  represents 
states  of  labile  equilibrium.  The  only  stable  forms  of  phosphorus 
which  have  been  observed  are  red  phosphorus  and  phosphorus 
vapor.  The  yellow  modification  and  the  melted  phosphorus  are 
both  labile  forms,  instable  in  respect  to  red  phosphorus.1  Whether 
the  curves  DC  and  ABmeet  at  some  higher  temperature  is  unknown. 
If  they  do  meet  there  can  coexist  at  that  point  red  phosphorus, 
liquid  and  vapor,  and,  from  there  on,  liquid  phosphorus  can  exist  in 
stable  equilibrium  with  the  vapor.  We  get  here  an  interesting  ap- 
plication of  the  statement  of  Reicher 2  that  the  reaction  velocity  is 
less  at  temperatures  far  below  the  inversion  point  than  in  its  imme- 
diate vicinity.  Above  520°  the  liquid  phosphorus  changes  into  the 


1  It  is  probable  that  the  substances  classified  as  "  monotropic  "  by  Lehman n 
(Molekularphysik  I,  193-219,)  are  analogous  to  phosphorus  ;  one  of  the  solid 
modifications  is  always  instable  with  respect  to  the  other  without  changing 
into  the  stable  form  very  rapidly. 

2Recueil  Trav.  Pays-Bas.  6,  251  (1883",;  Cf.  Gernez,  Couiptes  rendus, 
100,  1382  (1885). 


One  Component  33 

red  modification  so  rapidly  that  only  the  curve  AB  can  be  measured 
beyond  this  point.  Below  520°  it  is  possible  to  determine  the  values 
of  CD,  and  at  ordinary  temperatures  both  the  liquid  and  the  yellow 
phosphorus  are  fairly  permanent  even  in  the  presence  of  the  red 
variety.  There  is  no  difficulty  in  determining  the  temperature  of 
the  inversion  point  D,  44°,  although  the  non variant  system,  solid, 
liquid  and  vapor,  is  not  in  a  state  of  stable  equilibrium.  In  Table 
VI.  are  the  numerical  data  for  phosphorus.1  Under  A  are  the  vapor 
pressures  of  ordinary,  liquid  phosphorus ;  under  B  those  for  red 
phosphorus.  The  values  in  the  first  pressure-column  are  expressed 
in  millimeters  of  mercury  ;  in  the  other  two  in  atmospheres. 


TABLE  VI. 

Temperature 

A 

Temperature 

A 

B 

165° 

120  mm. 

360° 

3.2  Atm. 

0.6  Atm. 

170 

J73 

440 

7-5 

i  75 

180 

204 

4»7 

6.8 

200 

266 

494 

18.0 

20  > 

339 

503 

21.9 

219 

359 

510 

10.8 

226 

393 

5" 

26.2 

230 

5U 

531 

16.0 

290 

760 

550 

31.0 

577 

56.0 

The  relative  stability  of  labile  modifications  at  temperatures  far 
enough  below  the  inversion  temperature  is  further  illustrated  by  the 
behavior  of  arragonite  and  calcite,  two  modifications  of  calcium  car- 
bonate. On  heating,  the  former  changes  into  the  latter  ;  but  at 
ordinary  temperatures  arragonite  is  apparently  stable  even  in  con- 
tact with  calcite.1  The  apparent  stability  of  the  three  modifica- 
tions of  carbon,  and  of  titanic  acid, 3  is  doubtless  due  to  the  very 
high  inversion  temperatures  in  these  cases.  It  is  sometimes  thought 
that  the  occurrence  of  allotropic  modifications  is  something  unusual ; 


1  Schrotter,  Pogg.  Ann.  8 1,  276  (1850)  ;  Troost  and  Hautefeuille,  Comptes 
rendus,  76,  219  ( 1873). 

3  Rose,  Pogg.  Ann.  42,  360  (1837). 

3  Meyerhoffer,  Die  Phasenregel,  r8;  Lehmann,  Molekularphysik  I,  217. 
3 


34  The  Phase  Rule 

but  it  would  probably  be  quite  as  near  the  truth  to  say  that  most 
solids  can  exist  in  more  than  one  form,1  though  no  special  stress  has 
been  laid  upon  this  branch  of  the  subject  and  our  knowledge  of  the 
possible  modifications  of  many  compounds  is  very  rudimentary.  In 
Table  VII.  are  the  inversion  temperatures  of  a  few  of  the  substances 
that  have  already  been  studied/ 

The  coexistent  phases  in  all  these  cases  are  the  two  solid  modi- 
fications and  the  vapor. 

TABLE  VII. 


Mercuric  iodide  128.° 

Silver  iodide  146. 

Potassium  nitrate  I29-5 
Ammonium  nitrate  I  32.4 

Ammonium  nitrate  II  82.7 

Ammonium  nitrate  III  i25-5 

Silver  nitrate  J59-5 

Lead  nitrate  I  161.4 

Lead  nitrate  II  219.0 

Boracite  265.2 
Carbon  hexachloride  I  44. 

Carbon  hexachloride  II  71.1 

Carbon  tetrabromide  46.  i 


1  Cf.  Ivehmann,  Molekularphysik  I,  153-219  ;  D.  Berthelot,   Allotrople  des 
corps  simples  ;  Meslans,  Etats  allotropiques  des  corps  simples. 
2Schwarz>,  Prize  Dissertation,  Gottingen,  (1892). 


TWO   COMPONENTS 
CHAPTER  IV 

ANHYDROUS   SAI/T   AND   WATER 

A  phase  consisting  of  two  or  more  components  is  called  a  com- 
pound if  it  is  described  by  the  Theorem  of  Definite  and  Multiple 
Proportions,  a  solution  if  this  is  not  the  case.  A  solution  may  also 
be  defined  as  a  phase  in  which  the  relative  quantities  can  vary  con- 
tinuously within  certain  limits  or  as  a  phase  of  continuously  varying 
concentration.1  This  definition  does  not  confine  solutions  to  the 
liquid  phase,  but  includes  mixtures  of  gases  and  of  solids.  It  is  a 
question  whether  mixtures  of  gases  should  be  included.  In  so  far 
as  they  conform  to  the  Theorem  of  Dalton  they  are  only  mixtures 
and  the  variations  from  that  Theorem  are  scarcely  sufficient  to  per- 
mit one  to  classify  them  as  solutions.*  Since  gases  can  not  form  a 
surface  distinct  from  that  of  the  vessel  in  which  they  are  contained, 
they  are  therefore  consolute,5  i.  e.  miscible  in  all  proportions. 

The  conception  of  solid  solutions  is  due  to  van  't  Hoff.4  In- 
stances of  solid  solutions  are  to  be  found  in  such  salts  as  potash  and 
ammonia  alum,  beryllium  sulfate  and  seleniate,  ammonium  and 
ferric  chlorides,  potassium  and  thallium  chlorates  and  many  others.5 
The  distinction  between  isomorphic  solutions  and  mix  crystals  does 
not  seem  necessary,  the  only  difference  being  that  in  the  mix  crys- 
tals the  single  components  are  not  isomorphous.  Further  examples 
are  the  optically  homogeneous  colored  minerals  with  colorless  ground, 
the  glasses  and  some  alloys.  Under  the  same  head  are  probably  to 


1  Cf.  van  't  Hoff.  Zeit.  phvs.  Chem.  5,  323  (1890)  ;  Ostwald,  Lehrbuch  I, 
6o5 ;  Nernst,  Theor.  Chem.  87  ;  Le  Chatelier,  Equilibres  chimiques,  133. 

'Galitzine,  Wied.  Ann.  41,  588,  770  (1890). 

3  Bancroft,  Phys.  Rev.  3,  21  (1895). 

*Zeit.  phys.  Chem.  5,  322  (1890). 

'  Roozeboonj,  Ibid.  8,  530  (1891);  JO,  148(1892);  Fock,  Ibid.  ia,  657 
(1893)  ;  Stortenbeker,  Ibid.  16,  250  (1895). 


36  The  Phase  Rtile 

be  included  the  cases  of  occlusion '  by  precipitation  so  common  in 
analytical  chemistry  and  possibly  the  gelatinous  colloidal  hydrates. 
The  absorption  of  hydrogen  and  other  gases  by  metals  is  certainly 
due  in  part  to  the  formation  of  solid  solutions  as  is  also  the  absorp- 
tion of  oxygen  and  carbonic  acid  by  glass  at  a  temperature  of  200° 
under  200  Atm.  pressure.2  These  mixtures  show  some  of  the  prop- 
erties of  liquid  solutions  though  in  a  much  less  marked  manner, 
owing  to  the  resistance  to  change  due  to  the  solid  state.  Violle 3 
found  that  when  a  porcelain  crucible  was  heated  in  charcoal,  the 
carbon  diffused  through  it.  Carbon  diffuses  in  perceptible  quanti- 
ties into  an  iron  bar  during  one  day's  heating  at  250°. 4  Copper  has 
been  known  to  diffuse  into  platinum  and  into  zinc.5  Warburg  found 
that  it  was  possible  to  electrolyze  glass  between  electrodes  of  sodium 
amalgam.6  There  occurred  an  interesting  example  of  the  limitations 
introduced  •  by  the  solid  state.  It  was  possible  to  pass  sodium 
through  sodium  glass  without  any  change  being  visible.  If  elec- 
trodes of  lithium  amalgam  \vere  used,  the  sodium  was  replaced  by 
lithium  without  difficult}7 ;  but  the  glass  became  opaque  and  crumbly 
because  this  change  is  accompanied  by  contraction.  On  the  other 
hand,  since  the  replacement  of  sodium  by  potassium  involves  ex- 
pansion it  was  found  impossible  to  electrolyze  a  sodium  glass  be- 
tween electrodes  of  potassium  amalgam  while  there  was  no  difficulty 
when  a  potash  glass  was  substituted.  In  the  electrolysis  of  rock 
crystal  further  complications  were  introduced  with  the  crystalline 
form,  it  being  possible  for  the  current  to  pass  in  a  direction  parallel 
to  the  main  axis  and  impossible  in  the  direction  perpendicular  to  it.7 
In  a  solution  containing  two  components  only,  one  is  called  the 
solvent,  the  other  the  solute  or  dissolved  substance.6  In  cases  of 


'Schneider,  Ibid.  IO,  425  (1892). 

2Hannay,  Chetn.  News.  44,  3  (1881). 

sComptes  rendus,  94,  28  (1882)  ;  also  Marsden,  Proc.  Edinburgh  Soc.  IO, 
712  (1880). 

4Colson,  Comptes  rendus,  93,  1074  (1881). 

5Cf.  also  Roberts-Austen,  Phil.  Mag.  (5)  41,  526  (1896). 

6Wied.  Ann.  21,  622  (1884) ;  Tegettneier,  Ibid.  41,  18  (1890). 

7  Warburg  and  Tegetmeier,  Wied.  Ann.  35,  455  (1888)  ;  Tegetmeier,  Ibid. 
41,  18(1890). 

"Bancroft,  Proc.  Am.  Acad.  30,  324  (1894)  ;  Cf.  Story-Mankleyne,  Intro- 
duction to  Fock's  Chemical  Crystallography. 


Two  Components  37 

limited  miscibility  there  is  no  difficult}'  in  telling  which  component  is 
solvent  and  which  solute  ;  but  when  the  two  substances  are  conso- 
lute  there  is  at  present  no  sure  way  of  deciding  at  what  concentration 
the  change  takes  place.  As  this  is  essentially  a  quantitative  matter, 
it  does  not  affect  the  consideration  of  equilibrium  as  described  by 
the  Phase  Rule,  and  when  one  component  is  present  in  large  excess 
it  is  safe  to  speak  of  it  as  the  solvent.  This  distinction  between 
solvent  and  solute  does  not  apply  to  mixtures  of  gases,1  so  far  as  is 
yet  known,  an  additional  reason  for  not  considering  them  as  solu- 
tions. 

When  there  are  two  components  it  requires  four  coexisting 
phases  to  constitute  a  non variant  system,  three  and  two  for  a  mono- 
variant  and  a  di variant  system  respectively.  Since  no  two  compo- 
nents exhibit  all  the  types  of  equilibrium,  it  will  be  better  to  con- 
sider a  series  of  characteristic  pairs,  each  illustrating  some  new  case 
of  equilibrium  at  an  easily  accessible  temperature  and  pressure.  It 
will  then  be  possible  to  classify  the  different  phenomena  so  as  to  gain 
a  view  of  the  whole  field.  Having  studied  in  detail  the  effect  of 
changes  of  external  pressure  and  temperature  on  a  system  of  one 
component  it  will  not  be  necessary  to  repeat  this  when  there  are 
more  components  unless  there  is  some  new  feature  introduced  there- 
by. The  first  case  to  consider  is  the  one  where  the  two  components 
do  not  crystallize  together,  exist  each  in  only  one  solid  modification, 
and  there  is  only  one  nonvariant  system  possible  at  ordinary  tem- 
perature, two  solid  phases,  solution  and  vapor.  The  equilibrium 
between  potassium  chloride  and  water  will  serve  as  a  type.  The 
graphical  representation  of  this  system  with  the  pressure  and  tem- 
perature as  co-ordinates  is  given  approximately  in  Fig.  4. 

The  nonvariant  system,  potassium  chloride,  ice,  solution  and 
vapor,  is  found  to  be  possible  experimentally  at  one  temperature  and 
one  pressure  only,  represented  in  the  pressure-temperature  diagram 
by  the  point  O.  Any  continued  change  in  the  external  conditions 
produces  finally  the  disappearance  of  one  of  the  phases,  the  tem- 
perature and  pressure  remaining  constant  so  long  as  all  four  are 
present.  Which  of  the  two  solid  phases  disappears  first  on  addition 


1  The  distinction  undoubtedly  exists  in  many  cases  though  it  lacks  experi- 
mental confirmation. 


The  Phase  Rule 


FIG.  4. 

of  heat  depends  on  the  relative  quantities  of  the  two,  and  if  present 
in  the  same  proportion  as  in  the  solution  they  will  disappear  simul- 
taneously. In  this  case  the  whole  of  the  solid  will  melt  or  the  whole 
of  the  solution  will  freeze  without  change  of  temperature,  a  be- 
havior which  is  often  assumed  erroneously  to  be  a  criterion  of  the 
purity  of  a  compound.1  The  particular  mixtures  of  salts  and  ice 
which  have  a  constant  melting  point  were  called  cryohydrates  by 
Guthrie 2  and  were  supposed  to  be  compounds.  This  was  shown  not 
to  be  the  case  by  Pfaundler,3  Offer*  and  others5  for  the  following 
reasons  :  The  compositions  of  the  supposed  compounds  did  not  con- 
form to  the  Theorem  of  Definite  and  Multiple  Proportions  ;  the 


JCf.  Remsen,  Organic  Chemistry  7. 

2 Phil.  Mag.  (4)  49,  i,  206,  266  (1875) ;  (5)  r,  49,  354,  446  ;  2,  211  (1876) 
6,  135  (1878). 

3Ber.  chem.  Ges.  Berlin,  2O,  2223  (1877). 

4  Sitzunzsber.  Akad.  Wiss.  Wien,  8l,  II,  1058  (1880). 

5  It  is  interesting  to  note  that  the  true  explanation   was  offered  by  Schultz, 
Pogg.  Ann.  137,  247  (1869).  six  years  before  Guthrie's  first  paper. 


Components  39 

crystals  were  never  transparent  and  therefore  probably  not  homo- 
geneous ;  the  specific  volumes  and  the  heats  of  solution  were  addi- 
tive properties.  Another  reason  for  considering  the  crystals  as  in- 
homogeneous  mixtures  was  that  alcohol  dissolved  the  ice,  leaving 
the  salt ;  but  this  point  is  not  well  taken,  for  alcohol  will  dissolve 
cupric  chloride,  from  the  double  chlorides  of  copper  and  potassium.1 
From  the  point  of  view  of  the  Phase  Rule  it  is  clear  that  it  is  be- 
cause the  two  substances  do  not  crystallize  together  that  there  are 
four  phases  and  a  constant  freezing  point.  The  phenomenon  is  en- 
tirely general  and  occurs  in  all  cases  when  a  solution,  saturated  in 
respect  to  a  solid,  is  cooled  to  the  temperature  at  which  the  solvent 
begins  to  freeze  out.  The  cryohydric  temperature  is  the  temperature 
of  intersection  of  a  solubility  and  a  fusion  curve.*  The  easiest  way 
to  prepare  a  cryohydrate  is  to  cool  a  saturated  solution  till  the  tem- 
perature remains  constant,  pour  off  the  remaining  solution  and  let  it 
solidify,  which  it  will  do  without  change  of  temperature.  In  Table 
VIII.  are  the  cryohydric  temperatures  of  several  salt  solutions,  and 

TABLE  VIH 


Potassium  bromide 

-i3-° 

13-9 

Potassium  chloride 

-11.4 

16.6 

Potassium  iodide 

—  22. 

8-5 

Potassium  nitrate 

-    3-6 

44.6 

Potassium  sulfate 

—      1.2 

114.2 

Sodium  bromide 

—  24. 

8.1 

Sodium  chloride 

—  22. 

10.5 

Sodium  iodide 

—  15 

5-8 

Sodium  nitrate 

—  17-5 

8.1 

Sodium  sulfate 

—     0.7 

165-6 

Ammonium  bromide 

—  17- 

ii.  i 

Ammonium  chloride 

—  15- 

12-4 

Ammonium  iodide 

-27-5 

6.4 

Ammonium  nitrate 

—  17.2 

5-7 

Ammonium  sulfate 

—  17- 

IO.2 

1  Bancroft,  Phys.  Rev.  3,  401  (1896)  ;  Cf.  Ambronn  and  Le  Blanc,  Zcit. 
phys.  Chem.  16,  179  (1895) ;  Kfister,  Ibid.  525. 

1  Bancroft,  Jour.  Phys  Chem.  X,  No.  3  (1896).  This  definition  holds  only 
when  the  solution  is  saturated  in  respect  to  a  solid.  It  is  not  necessary  that 
what  separates  should  be  one  of  the  pure  components ;  it  may  contain  both 
components  as  in  the  case  of  a  hydrated  salt- 


40  The  Phase  Rule 

the  compositions  of  the  cryohydric  mixtures.  The  concentrations 
are  expressed  in  reacting  weights  of  water  per  reacting  weight  of 
salt.1 

If  solid  salt  be  present  in  excess,  the  ice  will  be  the  first  phase 
to  disappear  on  addition  of  heat,  leaving  the  monovariant  system, 
salt,  solution  and  vapor.  For  each  temperature  there  will  be  a  defi- 
nite pressure  in  the  vapor  phase  and  a  definite  concentration  in  the 
solution  at  which  the  system  can  be  in  equilibrium  and  this  pressure 
and  this  concentration  will  vary  with  the  temperature.  In  the  dia- 
gram, the  curve  OA  represents  the  pressures  and  temperatures  at 
which  the  saturated  solution  can  exist.  It  is  a  solubility  curve, 
water  being  the  solvent.  Addition  of  liquid  to  the  solution  from 
outside  or  from  the  vapor  phase  by  condensation  causes  more  of  the 
salt  to  go  into  solution  until  the  equilibrium  concentration  is  restored  ; 
removal  of  water,  by  evaporation  for  instance,  brings  about  a  pre- 
cipitation of  the  solute.  Both  these  changes  are  in  accordance  with 
the  Theorem  of  L,e  Chatelier.  Addition  of  water  means  a  decrease 
in  the  concentration  of  the  salt  which  is  neutralized  by  more  salt 
going  into  solution.  Removal  of  water  increases  the  concentration 
and  the  equilibrium  is  restored  by  elimination  of  the  excess  of  salt. 
Since  it  is  easier  to  measure  concentrations  than  vapor  pressures  it  is 
more  familiar  to  every  one  that  for  each  temperature  there  is  a  single 
well-defined  solubility  than  that  the  same  is  also  true  for  the  pres- 
sures. In  one  case  it  is  easy  to  show  the  applicability  of  the  Phase 
Rule  to  the  relation  between  pressure  and  temperature.  There 
should  be  but  one  temperature  at  which  the  vapor  pressure  of  a 
monovariant  system  can  be  equal  to  the  atmospheric  pressure,  and  it 
is  found  experimentally  that  the  boiling  point  of  a  saturated  solu- 
tion is  constant  so  long  as  the  three  phases  are  present  and  the 
barometric  pressure  remains  unaltered.2  The  concentration  of  the 
solution  changes  with  the  temperature  and  the  direction  of  this 
change  can  be  foretold  from  the  Theorem  of  I^e  Chatelier.  If  the 
solid  dissolves  with  absorption  of  heat,  it  will  dissolve  in  greater 


'Guthrie,  Phil.  Mag.  (4)  49,  269  (1875). 

-  For  boiling  points  of  saturated  salt  solutions,  see  Landolt  and  Bernstein's 
Tabelleu,  232. 


Components  41 

quantity  if  the  temperature  of  the  system  rises,  as  this  change  in- 
volves addition  of  heat.1 

Since  most  salts  dissolve  in  water  with  absorption  of  heat,  the 
increasing  solubility  with  rising  temperature  is  just  what  one  would 
have  expected.  There  are  substances  known,  such  as  calcium  hy- 
drate, sodium,  cerium  and  thorium  sulfates,1  and  calcium  isobutyrate 
which  evolve  heat  on  going  into  solution,  and,  in  all  these  cases, 
there  is  decreasing  solubility  with  increasing  temperature.*  As  it  is 
not  necessary  that  the  heat  of  solutiou  should  have  the  same  sign  at 
all  temperatures,  it  is  possible  for  the  solubility  of  a  salt  to  increase 
with  rising  temperature  and  then  decrease  as  the  temperature  rises 
still  higher  or  vice-versa.  Examples  of  the  first  type  are  calcium 
sulfate  which  reaches  a  maximum  solubility  between  thirty  and  forty 
degrees,*  and  calcium  isobutyrate  which  has  a  maximum  solubility 
in  the  neighborhood  of  8o°.5  Whether  the  decreasing  solubilities  of 
the  many  sulfates,  sulfites,  oxalates  and  carbonates  at  high  tempera- 
tures are  further  illustrations  of  this  is  not  certain,  as  it  has  not  been 
shown  that  the  same  substance  crystallizes  from  the  solutions  at  the 
different  temperatures.*  An  example  of  the  second  class  where  the 
solubility  decreases  at  first  to  increase  later  is  to  be  found  in  calcium 
butyrate  and  possibly  in  sodium  sulfate.  This  last  is  said  to  have  a 
minimum  solubility  at  about  I25°.7  In  all  these  cases  the  heat  of 
solution  is  zero  at  the  temperature  of  the  maximum  or  minimum 
solubility,  whichever  it  happens  to  be.  There  is  no  instance  known 
where  the  sign  of  the  heat  effect  is  not  in  accordance  with  the  The- 
orem of  Le  Chatelier  though  the  contrary  has  been  maintained 
owing  to  a  false  application  of  the  theorem/  The  theorem  predicts 
the  direction  of  the  change  when  the  system  passes  from  one  state 
of  equilibrium  to  another  owing  to  a  change  in  one  or  more  of  the 
factors  of  equilibrium.  In  the  particular  case  in  hand,  the  change 


1  Le  Chatelier,  Eqnilibres  chimiqaes,  50. 

•  I  am  told  that  this  is  characteristic  of  the  sulfates  of  all  the  rare  earths. 
*Cf.  Roozeboom,  Recueil  Trav.  Pays- Has,  8,  137  (1889). 

4  Berthelot,  Mecanique  chimique  I,  131. 

J  Le  Chatelier,  Comptes  rendus,  104,  679  (1887). 

«Etard,  Ibid.  106,  206,  740  (1888). 

•  Tilden  and  Shenstone.  Phil.  Trans.  175,  23  (1884). 
"Chancel  and  Parmentier,  Comptes  rendns.  104,  474  (1887). 


42  The  Phase  Rule 

of  solubility  with  the  temperature  can  be  foretold  from  the  heat 
evolved  or  absorbed  when  we  pass  from  one  saturated  solution  to 
another.  To  put  it  differently,  the  important  point  is  the  sign  of  the 
heat  effect  when  the  solute  is  added  to  an  almost  saturated  solution. 
It  is  not  proper  to  consider  the  heat  evolved  or  absorbed  when  the 
solute  is  added  to  pure  water.  This  last  is  the  heat  of  solution 
usually  determined  in  thermochemistry  because  it  is  easier  to 
measure.  The  heat  of  precipitation,  on  the  other  hand,  is  very 
nearly  the  heat  referred  to  in  the  Theorem  of  L,e  Chatelier.  While 
the  heat  of  solution  in  the  thermochemical  use  of  the  term  or  the 
heat  effect  when  the  solute  is  dissolved  in  much  water  has  usually 
the  same  sign  as  the  negative  heat  of  precipitation  this  is  by  no 
means  always  the  case.  Calcium  isobutyrate,  at  temperatures  below 
80°,  dissolves  in  a  great  deal  of  water  with  evolution  of  heat  ;  in  a 
little  water  with  absorption  of  heat.  Reicher  and  van  Deventer 
showed  that  the  same  thing  took  place  with  cupric  chloride,  there 
being  an  evolution  of  heat  when  the  salt  dissolved  in  a  large  excess 
of  water,  and  also  an  evolution  of  heat  when  the  salt  was  precipi- 
tated from  a  supersaturated  solution.1  It  follows  from  this  behavior 
that  there  must  be  some  quantity  of  water  in  which  one  gram  of  salt 
will  dissolve  withont  either  evolution  or  absorption  of  heat.  This 
conclusion,  which  has  no  theoretical  importance,  was  verified  ex- 
perimentally. The  same  phenomenon  has  been  observed  with  the 
hydrates  of  ferric  chloride.2  When  the  heat  of  precipitation  of  a 
solute  is  zero  the  solubility  does  not  change  with  the  temperature. 
This  is  very  nearly  realized  in  the  case  of  sodium  chloride.  The 
absolute  solubilities  of  different  solids  in  liquid  solvents  is  a  subject 
about  which  it  is  impossible  to  make  any  predictions  in  the  present 
state  of  our  knowledge.  There  are  all  degrees  of  miscibility  from 
barium  sulfate  which  is  .soluble  approximately  one  part  in  four 
hundred  thousand  of  water  to  pyrogallol  which  is  miscible  in  nearly 
all  proportions  with  water.  Save  for  a  few  empirical  generalizations 
our  ignorance  is  complete.3 

'Zeit.  phys.  Chem.  5,  559  (1890). 
2Roozeboom,  Ibid.  IO,  501  (1892). 

3  Cf.  Ostwald,    Lehrbuch  I,    1066 ;  Carnelley,  Jour.    Chetn.    Soc.   53,  782 
(1888)  ;  Etard,  Comptes  rendus,  98,  1276  (1884)  ;  Vaubel,  Jour,  prakt.  Chem. 

(2)52,72(1895). 


Tu-o  Components  43 

The  vapor  pressure  of  the  monovariant  system,  salt,  solution 
and  vapor  is  always  less  than  that  of  the  pure  solvent  at  the  same 
temperature  provided,  as  in  this  case,  the  vapor  pressure  of  the 
solute  can  be  disregarded.  If  the  solute  has  a  perceptible  vapor 
pressure  of  its  own,  the  vapor  pressure  of  the  system  may  be  greater 
or  less  than  that  of  either  component  when  pure  ;  but  the  partial 
pressure  of  the  solvent  in  the  vapor  space  above  the  liquid  is  always 
less  than  its  pressure  in  the  pure  state.  That  this  must  be  so  can 
be  seen  by  an  application  of  the  Theorem  of  Le  Chatelier.1  Sup- 
pose we  have  a  liquid  in  equilibrium  with  its  own  vapor  and  add  a 
small  quantity  of  some  substance  soluble  in  the  liquid.  There  will 
be  a  tendency  to  eliminate  the  disturbing  factor  by  condensation  of 
vapor,  thus  reducing  its  concentration.  This  change  will  take  place 
until  equilibrium  is  reached  at  a  diminished  vapor  pressure  for  the 
solvent.  This  reasoning  does  not  apply  to  the  solute  for  its  concen- 
tration might  be  diminished  by  increasing  its  volatility.  No  cases 
of  this  sort  have  been  studied  quantitatively  as  yet.  though  the 
theory  of  distillation  with  steam  can  not  be  worked  out  compietely 
until  this  is  done.  As  more  of  the  solute  is  added,  the  vapor  pres- 
sure of  the  solvent  decreases  until  the  maximum  concentration  or 
point  of  saturation  of  the  liquid  phase  is  reached,  when  the  disturb- 
ing factor  is  eliminated  by  precipitation,  the  vapor  pressure  of  the 
system  remaining  constant.  It  follows  that  the  boiling  point  of  a 
solution  saturated  in  respect  to  a  non-volatile  solute  will  always  be 
higher  than  that  of  the  pure  solvent.  The  vapor  pressure  of  the 
monovariant  system,  salt,  solution  and  vapor,  increases  with  rising 
temperature  but  not  so  rapidly  as  that  of  the  pure  solvent,  because 
of  the  ever  greater  depression  due  to  increasing  solubility.  It  is 
conceivable,  theoretically,  that  the  lowering  of  the  vapor  pressure 
due  to  increased  solubility  might  be  so  great  as  more  than  to  coun- 
terbalance the  normal  increase  conditioned  by  the  heat  of  vaporiza- 
tion, in  which  case  there  would  be  a  decrease  of  pressure  with  in- 
creasing temperature.1  This  has  been  realized  in  the  case  of  calcium 
chloride.*  It  should  be  kept  in  mind  that  there  is  nothing  in  the 

2  Bancroft,  Jour.  Phys.  Chem.  I,  No.  3  (1896). 

-Meyerhoffer  (Die   Phasenregel,   25.)  has  made  his  diagram   as  if  this 


'  Roazeboom.  Zeit  phys.  Chem.  4,  45  (1889). 


44  The  Phase  Rule 

Phase  Rule  to  imply  that  the  heat  of  vaporization  of  a  salt  solution 
is  the  same  as  that  of  the  pure  solvent.  This  assumption  is  in- 
volved in  the  Theorem  of  v.  Babo1  and  in  most  of  the  modern 
quantitative  work  on  vapor  pressures,  but  it  is  probably  not  accurate 
in  any  case.  That  it  is  a  very  close  approximation  in  many  in- 
stances is  shown  by  Raoult's  work  on  vapor  pressures  of  dilute  solu- 
tions.2 As  the  temperature  rises  the  vapor  pressure  of  the  saturated 
solution  becomes  greater  up  to  the  critical  point  of  the  solution  be- 
yond which  temperature  and  pressure,  represented  in  the  diagram 
(Fig.  4)  by  A,  this  monovariant  system  can  no  longer  exist.  There 
are  four  possible  cases  each  of  which  seem  to  have  been  observed 
experimentally.  The  solid  may  melt  and  not  be  consolute  with  the 
solution.  This  occurs  with  naphthalene  and  water,  sulfur  and 
toluene  for  instance  ;  the  point  A  is  a  quadruple  point,  the  non- 
variant  system  being  composed  of  a  solid,  a  vapor  and  two  liquid 
phases.  The  solid  may  melt  and  mix  with  the  solution,3  leav-' 
ing  two  phases,  solution  and  vapor.  An  instance  of  this  is  silver 
nitrate  and  water.*  The  solubility  may  decrease  until  the  solution 
and  vapor  have  the  same  composition  and  there  is  left  the  solid  and 
a  phase  which  is  either  liquid  or  vapor  as  one  chooses.  This  seems 
to  occur  in  solutions  of  sulfates  in  water  though  the  experiments 
have  not  been  pushed  to  the  critical  point.5  Lastly  the  solution  and 
vapor  may  come  to  have  the  same  composition  by  increased  vapor- 
ization of  the  solute  instead  of  by  decreased  solubility  as  in  the  pre- 
ceding case.  This  has  been  realized  by  Hannay 6  with  potassium 
iodide  and  alcohol.  He  did  not  work  with  a  saturated  solution,  but 
an  increase  in  concentration  would  not  change  the  character  of  the 
phenomenon,  only  the  temperature  at  which  it  takes  place.  In  this 
last  case, 'the  critical  temperature  of  the  solution  lies  between  the 


1  Ostwald,  Lehrbuch  I,  706. 

2  Comptes  rendus,  XOJ,  1125  (1886);  Cf.  also,  Emden,  Wied.  Ann.  31,  145 
(1887)  ;  Dieterici,  Ibid.  42,  528  (1891). 

3  The  curve  at  this  point  is  a  fusion  and  no  longer  a  solubility  curve.     This 
will  be  considered  in  detail  later. 

4Etard,  Comptes  rendus,  108,  176  (1889). 
5  Etard,  Ibid.  106,  206,  740  (1888). 
6Proc.  Roy.  Soc.  30,  178  (1880). 


Tu-o  Components  45 

critical  temperatures  of  the  pure  components.1  In  the  other  direc- 
tion the  solubility  curve  has  been  followed  beyond  the  cryohydric 
temperature,*  but  the  system  is  then  in  a  state  of  labile  equilibrium 
and  the  addition  of  the  merest  fragment  of  an  ice  crystal  produces 
a  change  to  a  state  of  stable  equilibrium,  the  temperature  rising  to 
the  inversion  temperature. 

Starting  from  the  non variant  system,  salt,  ice,  solution  and 
vapor,  we  can  pass  to  the  monovariant  system,  ice,  solution  and  vapor, 
by  supplying  heat,  provided  ice  is  present  in  excess.  Since  the 
curve  OB.  representing  this  series  of  equilibria,  ends  at  the  melting 
point  of  pure  ice,  it  is  called  a  fusion  curve.  The  distinction  be- 
tween a  fusion  and  a  solubility  curve  is  that  the  former  ends  at  the 
melting  point  of  one  of  the  pure  components  while  the  latter  does 
not.  For  each  concentration  or  for  each  pressure,  which  is  another 
way  of  saying  the  same  thing,  there  is  but  one  temperature  at  which 
ice  can  be  in  equilibrium  with  solution  and  vapor.  It  is  usually 
assumed  that  these  three  phases  will  be  in  equilibrium  when  the 
vapor  pressure  of  the  solid  solvent  is  equal  to  the  partial  pressure  of 
that  component  in  the  system,  solution  and  vapor,*  but  this  is  an 
assumption  which  is  probably  never  accurately  true.  It  is  very 
doubtful  in  my  mind  whether  a  solution  of  alcohol  in  water  is  in 
equilibrium  with  ice  at  the  temperature  at  which  the  vapor  pressure 
of  the  ice  equals  the  partial  pressure  of  the  water  vapor  above  the 
solution.  This  difference  of  pressure  is  probably  very  slight,  in 
most  cases,  and  its  non-existence  has  seemed  so  self-evident  as  to 
require  no  proof.  The  vapor  pressures  of  the  fusion  curve  exceed 
those  of  the  solid  solvent  by  the  partial  pressure  of  the  solute  and 
the  correction  term  just  referred  to.  For  a  non-volatile  solute,  a 
solid  in  the  pure  state  at  the  temperature  of  the  experiment,  both  of 
these  variations  may  be  neglected  in  the  present  state  of  our  knowl- 
edge and  the  two  curves  are  identical.  Under  these  circumstances 
the  fusion  curve  OB  has  a  greater  vapor  pressure  for  a  given  tem- 
perature than  the  solubility  curve  OA,  because  the  solution  is  more 
dilute  in  the  first  case.  Since  the  solute  lowers  the  vapor  pressure 


1  Winkelmann,  Handbnch  der  Physik  II,  2,  668. 
'Guthrie,  Phil.  Mag.  (4^1  49,  214  (1875). 
*  Goldberg,  Comptes  rendus,  70,  1349(1870). 


The  Phase  Rule 


of  the  solution,  it  lowers  the  temperature  at  which  the  solution  is  in 
equilibrium  with  the  solid  solvent.  This  is  best  seen  from  the  dia- 
gram (Fig.  5). 


FIG.  5. 

OA  is  the  pressure-temperature  curve  for  the  liquid  solvent  in 
presence  of  vapor,  OB  the  corresponding  curve  for  the  solid  solvent 
and  O  the  freezing  point.  CC,  DD,  EE:  are  the  pressure  tempera- 
ture curves  for  solutions  of  ever  greater  concentration.  These  solu- 
tions freeze  at  the  temperatures  at  which  their  vapor  pressure  curves 
cut  the  curve  OB,  namely  at  C,  D  and  E.  The  presence  of  a  dis- 
solved substance  lowers  the  freezing  point  of  the  solution  in  all 
cases  where  pure  solid  solvent  separates  out.  Later  we  shall  see 
that  when  this  last  condition  is  not  fulfilled,  the  freezing  point  is  not 
necessarily  lowered. 

The  equilibrium  between  ice,  potassium  chloride  and  vapor  is 
represented  in  Fig.  4  by  the  curve  OC.  Any  change  in  external 
pressure  produces  a  change  in  the  quantity  of  one  of  the  phases  and 
for  each  temperature  there  is  but  one  pressure  at  which  the  system 
can  exist.  The  pressure  of  this  system  will  be  equal  to  the  sum  of 
the  vapor  pressures  of  the  two  solid  components  if  each  vapor  may 
be  considered  as  an  indifferent  gas,1  an  assumption  which  is  never 

1  Nernst,  Theor.  Chem.  376. 


Two  Components  47 

P 
absolutely  accurate.1     As  the  diagram  shows,  this  system  can  be  in 

equilibrium  only  at  temperatures  below  that  of  the  cryohydric 
point.  If  a  salt  is  mixed  with  ice  at  a  higher  temperature,  the  ice 
will  melt  and  the  salt  dissolve  until  one  or  the  other  disappears, 
forming  one  of  the  two  stable  mouovariant  systems,  salt,  solution 
and  vapor,  or  ice,  solution  and  vapor,  as  the  salt  or  ice  is  in  excess. 
The  ice  in  melting  absorbs  heat,  the  salt  in  dissolving  may  evolve  or 
absorb  heat,  if  the  latter,  as  is  usually  the  case,  both  these  changes 
work  in  the  same  direction  and  the  total  absorption  of  heat  is  very 
considerable.  It  is  this  phenomenon  which  is  utilized  in  many  arti- 
ficial freezing  mixtures.  The  system,  salt,  ice  and  vapor,  not  being 
in  equilibrium  at  the  temperature  of  the  experiment  tends  to  pass 
into  a  state  of  stable  equilibrium.  The  temperature  falls  until  one 
or  the  other  solid  phase  disappears  or  until  the  temperature  of  the 
cryohydrate  is  reached.  This  is  the  lowest  temperature  which  can 
be  attained  at  atmospheric  pressure  with  a  given  freezing  mixture, 
because  at  and  below  this  temperature  the  two  solid  phases  can  be  in 
equilibrium  with  each  other  and  will  not  react.  It  would  be  possi- 
ble by  keeping  the  mixture  in  an  air  current  to  obtain  slightly  lower 
temperatures,  but  the  cooling  in  this  case  would  be  due  to  the  heat 
absorbed  in  the  evaporation  of  the  ice  and  would  have  nothing  to  do 
with  the  nature  of  the  other  phase.  The  temperature  reached  in 
laboratory  experiments  is  by  no  means  always  that  of  the  cryohy- 
drate. The  two  components  are  mixed,  let  us  say,  at  zero  degrees 
and  the  temperature  at  once  falls.  At  the  same  time  there  is  an 
ever-increasing  amount  of  solution  formed  which  has  to  be  cooled 
down  as  well  as  the  rest  of  the  system.  The  result  is  that  there  is 
reached  a  point  where  the  heat  absorbed  by  the  amount  of  solution 
formed  in  the  unit  of  time  is  just  sufficient  to  keep  the  mass  of  solu- 
tion already  present  at  constant  temperature.  What  this  tempera- 
ture will  be  depends  on  the  initial  temperature,  on  the  rate  of  radia- 
tion, on  the  quantity  of  salt  and  ice  used  and  on  the  thoroughness 
with  which  the  ice  and  salt  have  been  mixed.  Since  the  reaction 
velocity  is  proportional,  among  other  things,  to  the  surfaces  of  the 
solid  phases  in  contact,  less  heat  will  be  absorbed  in  the  unit  of  time 
and  therefore  equilibrium  will  be  reached  at  a  higher  temperature 
'Bancroft,  Phys.  Rev.  3,  414  (1896). 


48  The  Phase  Ride 

when  the  ice  is  in  large  pieces  than  when  the  two  components  are 
ground  very  fine  and  intimately  mixed.  For  this  reason  snow  is 
better  than  ice,  being  more  finely  divided.  The  most  effective  ar- 
rangement for  a  salt-ice  freezing  mixture  is  to  have  the  mixture  in  a 
perforated  vessel  so  that  the  solution  can  run  off  as  fast  as  formed. 
When  the  temperature  has  fallen  to  the  cryohydric  temperature  the 
mixture  can  be  placed  in  an  impermeable  vessel  and  used  for  freez- 
ing purposes.  It  is  also  economical  to  take  the  two  components  in 
the  proportion  in  which  they  occur  in  the  solution  at  the  cryohydric 
temperature,  as  the  melting  point  of  the  cooled  solids  will  remain 
constant.1  It  is  clear  that  the  requisites  for  a  .successful  freezing 
mixture  are  that  the  cryohydric  temperature  should  be  very  low  and 
that  the  heat  absorbed  per  gram  of  solution  formed  should  be  as 
great  as  possible.  The  first  criterion  is  satisfied  if  the  salt  is  very 
soluble,  the  second  if  the  increase  of  the  solubility  with  the  temper- 
ture  is  great.  For  this  reason  the  freezing  mixture,  made  of  com- 
mon salt  and  ice,  so  often  used  in  the  laboratory,  is  not  a  good  one, 
except  on  the  ground  that  sodium  chloride  is  cheap.  As  the  heat  of 
solution  of  this  salt  is  very  slight  the  heat  absorbed  is  little  more 
than  the  heat  of  fusion  of  ice  alone.  A  much  better  mixture  is 
ammonium  nitrate  and  ice  and  even  better,  crystallized  calcium 
chloride.  The  anhydrous  salt  can  not  be  used  as  it  evolves  heat  in 
taking  up  six  units  of  water.  It  will  be  shown  later  that  this  is 
characteristic  of  all  hydrates.  While  the  freezing  mixtures  in  use 
are  often  composed  of  ice  and  some  salt,  this  is  not  essential.  An3' 
system  which  is  in  instable  equilibrium  and  which  absorbs  heat  in 
passing  into  the  stable  form  can  be  used  as  a  freezing  mixture,  such 
as  alcohol  or  sulfuric  acid  and  snow,  solid  carbonic  acid  and  ether, 
liquid  air  at  atmospheric  pressure. 

In  all  cases  where  two  substances  can  form  a  solution  and  can 
crystallize  from  that  solution  in  the  pure  state,  the  temperature  at 
which  the  two  components  can  be  in  equilibrium  with  the  solution  is 
lower  than  the  fusion  point  of  either  of  the  pure  components.  This 
is  a  necessary  consequence  of  the  Phase  Rule  and  the  Theorem  of 
L,e  Chatelier,  but  it  has  been  entirely  overlooked  by  Ivtard  in  some 

xData  for  many  salts  with  ice  can  be  found  in  the  papers  of  Guthrie  al- 
ready cited.     Cf.  also  Landolt  and  Bornstein's  Tabellen,  p.  315. 


Two  Components  49 

theoretical  views  advanced  by  him.1  He  observed  that  many  sub- 
stances were  very  slightly  soluble  in  water.  Taking  this  with  the 
fact  that  most  solubilities  decrease  with  falling  temperature,  he  drew 
the  conclusion  that  the  solubility  of  any  substance  at  the  freezing 
point  of  the  solvent  is  zero  if  the  solvent  freeze  at  a  sufficiently  low 
temperature.  This  is  wrong.  The  temperature  at  which  the  non- 
variant  system,  both  solids,  solution  and  vapor,  exists  is  the  limiting 
point  for  the  stable  monovariant  system,  solid  solute,  solution  and 
vapor,  and  while  it  ma}-  in  some  cases  approach  infinitely  near  to 
the  fusion  point  of  the  solvent  it  can  never  reach  it.  Arctowski  * 
has  published  some  measurements  showing  the  inaccuracy  of  Etard's 
hypothesis  but  seems  to  have  equally  erroneous  ideas  in  regard  to 
what  actually  happens. 

The  fourth  monovariant  system,  represented  by  the  curve  OD,  is 
the  one  in  which  there  is  equilibrium  between  salt,  ice  and  solution. 
The  direction  of  this  curve  is  determined  by  the  difference  between 
the  sum  of  the  volumes  of  the  solid  solvent  and  solute  and  the  vol- 
ume of  the  same  masses  as  solution.  It  is  also  dependent  on  the 
sign  of  the  heat  effect  when  the  two  solid  phases  pass  into  solution. 
If  the  solution  is  formed  with  expansion  of  volume  and  absorption 
of  heat  or  with  contraction  of  volume  and  evolution  of  heat,  in- 
crease of  pressure  will  raise  the  temperature  at  which  the  two  solid 
phases  can  be  in  equilibrium  with  the  solution.  If  the  solution  is 
formed  with  expansion  of  volume  and  evolution  of  heat,  or  con- 
traction of  volume  and  absorption  of  heat,  the  curve  OD  will  slant  to 
the  left  as  it  is  drawn  in  the  diagram  instead  of  to  the  right,  show- 
ing that  increase  of  pressure  lowers  the  freezing  point  of  the  solu- 
tion. The  only  difference  between  this  case  and  that  where  there  is 
only  one  component  is  that  the  change  from  pure  solid  to  pure  liquid 
is  always  accompanied  by  an  absorption  of  heat,  while  it  is  conceiv- 
able that  two  substances  might  exist  such  that  the  heat  of  solution 
evolved  would  be  greater  than  the  heat  of  fusion  of  the  other  sub- 
stance. No  instance  of  this  is  on  record  to  the  best  of  my  knowl- 
edge. The  Theorem  of  Le  Chatelier  enables  us  to  predict  that  in 
such  a  case  the  solid  mass  would  liquefy  on  cooling. 

1  Ann.  chim..phys.  (7)  «,  268  <  1894). 
-  Comptes  rendus,  xai,  123  (1895). 
4 


5O  The  Phase  Rule 

With  water  as  solvent,  the  most  usual  case  is  that  the  volume  of 
the  solution  is  less  than  that  of  the  two  solid  components,  while  the 
reverse  is  true  for  most  saturated  solutions  in  other  solvents.  Little 
is  known  experimentally  of  the  course  of  this  curve,1  but  it  seems 
probable  that  it  will  either  approach  asymptotically  the  pressure  of 
the  system,  solid  and  liquid  solvent,  or  that  it  will  terminate  at  the 
intersection  with  the  curve  for  solid  solute,  solid  solvent  and  solution 
provided  the  vapor  pressure  of  the  solute  may  be  neglected. 

The  different  divariant  systems  exist  in  the  fields  limited  by  the 
boundary  curves  and  by  the  curves  for  the  pure  components.  The 
system,  unsaturated  solution  and  vapor,  can  exist  at  a  series  of  tem- 
peratures and  for  each  temperature  at  a  series  of  pressures  depend- 
ing on  the  concentrations.  If  the  temperature  and  concentration 
are  both  fixed,  the  pressure  is  determined  thereby.  If  the  solution 
is  heated  in  an  open  vessel  so  that  the  solvent  can  distill  off,  the 
boiling  point  will  rise  with  increasing  concentration  until  the  solu- 
tion becomes  saturated  ;  after  which  it  remains  constant.  The  vapor 
pressure  of  a  dilute  solution  is  less  than  that  of  the  pure  solvent  if 
the  partial  pressure  of  the  solute'  may  be  neglected.2  For  all  prac- 
tical purposes,  potassium  chloride  conies  under  this  head  and  its 
vapor  pressure  may  be  considered  equal  to  zero.3  I  shall  discuss  the 
limiting  pressure  and  temperature  for  the  possible  divariant  systems 
on  this  assumption,  showing  afterwards  the  changes  necessitated  by 
dropping  it.  For  purposes  of  reference  I  have  added  to  the  diagram 
( Fig.  4)  the  curves  BBj  and  BD,  which  represent  the  equilibrium 
between  water  and  vapor,  water  and  ice  respectively.  The  curve  for 
ice  and  vapor  coincides  with  the  curves  BO  and  OC,  so  far  as  we 
know  now,  when  we  neglect  the  vapor  pressure  of  the  potassium 
chloride.  All  possible  temperatures  and  pressures  at  which  there 
can  be  equilibrium  between  solution  and  vapor  lie  within  the  space 
AOBB,.  The  points  Ei  and  A,  the  critical  points  of  the  solvent  and 
solution  respectively,  will  coincide  only  if  the  solubility  of  the  solute 


1  Roloff,  Zeit.  phys.  Chem.  X7,  348  (1895). 

2  If  we  eliminate  the  natural  and  the  induced  vapor  pressure  of  the  solute 
as  well  as  the  effect  due  to  surface  tension,   BOC  becomes  a  continuous  curve 
with  no  change  of  direction  at  O. 

*This  is  not  strictly  true.     Cf.  Bailey,  Jour.  Ghem  Soc.  65,  445  (1894). 


Two   Components  51 

becomes  zero  at  the  critical  temperature  and  pressure  for  the  solvent. 
It  has  already  been  pointed  out  that  this  seems  to  occur  very  nearly 
for  many  sulfates,  sulfites,  carbonates  and  oxalates1  ;  but  this  is  not 
a  general  phenomenon,  and  even  in  these  instances  the  conclusion  is 
based  on  extrapolation  in  an  empirical  formula  and  not  on  experi- 
mental data.  In  the  field  bounded  by  the  line  AOC  and  the  temper- 
ature axis  there  is  equilibrium  between  salt  and  water  vapor  ;  and 
here,  too,  the  equilibrium  is  not  settled  definitely  till  two  variables 
are  determined  arbitrarily,  the  pressure  and  the  temperature  or  one 
of  these  and  the  concentration.  Practical!}',  the  only  change  of 
concentration  is  in  the  vapor  phase  ;  but,  theoretically,  the  density 
of  the  solid  phase,  and  therefore  its  volume  concentration,  must 
change  with  changing  pressure  or  temperature.  If  the  pressure  is 
increased  above  that  of  the  saturated  solution,  represented  by  the 
curve  OA,  vapor  will  condense,  forming  a  saturated  solution.  In 
other  words,  an  anhydrous  salt  is  deliquescent  when  the  pressure  of 
water  vapor  in  the  atmosphere  is  greater  than  the  vapor  pressure  of 
the  saturated  solution  and  is  permanent  when  it  is  less.  If  instead 
of  diminishing  the  pressure  upon  a  saturated  solution  reaching  the 
di variant  system,  solid  solute  and  vapor,  we  increase  it  there  is 
formed  the  system,  solution  and  vapor,  if  there  is  only  a  small 
amount  of  undissolved  solid  and  a  large  amount  of  vapor,  or  the 
system,  salt  and  solution,  if  there  is  an  excess  of  the  solid  phase. 
This  new  divariant  system,  existing  only  in  the  field  AOD,  can 
have  different  concentrations  and  pressures  at  the  same  temperature  ; 
but  there  is  a  definite  solubility  for  each  pressure  at  each  tempera- 
ture. The  change  in  the  concentration  of  the  solution  is  in  the 
direction  predicted  by  the  Theorem  of  Le  Chatelier.  If  the  volume 
of  the  solid  plus  the  volume  of  the  solvent  is  greater  than  the  vol- 
ume of  the  resulting  solution,  increasing  pressure  means  increasing 
solubility,  otherwise  it  involves  decreasing  solubility,  the  system 
being  always  kept  at  constant  temperature.*  With  most  salts  there 


1  Etard,  Comptes  rendus,  106,  206,  740  (1888)  ;  Ann.  chim.  phys.  (7)  2, 
546(1894). 

2Braun,  Wied.  Ann.  30,  250  (1887);  v.  Stackelberg,  Zeit.  phys.  Cheni. 
2O,  337  (1896)  ;  Ostwald,  Lehrbuch  I,  1044-1047.  It  is  incorrect  to  say  that  the 
change  in  the  solubility  depends  also  -on  the  sign  of  the  heat  of  solution.  In 


52  The  Phase  Rule 

is  a  decrease  of  volume  when  dissolving  in  water  which  is  so  great 
in  the  case  of  copper  sulfate  that  a  dilute  solution  of  that  salt  occu- 
pies less  volume  than  the  pure  solvent  alone.1  There  are  cases 
known  where  there  is  an  increase  of  volume  when  the  salt  goes  into 
solution,  the  most  notable  instance  being  ammonium  chloride.  In 
this  case,  increase  of  pressure  involves  precipitation  of  the  salt. 

In  the  field  DBOD,  there  can  exist  the  system,  salt  and  solution, 
as  we  have  just  seen  and,  under  proper  conditions,  the  system,  ice 
and  solution.  This  divariant  system  has  been  very  little  studied, 
the  only  interesting  investigation  of  the  subject  being  a  paper  by 
Colson.2  He  showed  that  if  the  concentration  and  temperature  be 
fixed  the  pressure  has  a  definite  value  which  is,  of  course,  a  neces- 
sary consequence  of  the  Phase  Rule.  In  the  field  DOC  there  is 
equilibrium  between  salt  and  ice.  The  divariant  system,  ice  and 
vapor,  can  not  be  realized  if  the  vapor  pressure  of  potassium 
chloride  be  treated  as  equal  to  zero.  If  this  assumption  be  given 
up,  the  curve  for  the  equilibrium  between  ice  and  its  own  vapor 
will  no  longer  coincide  with  BOC,  but  will  be  represented  by  the 
line  BHC.  At  the  same  time  it  will  be  necessar}7  to  add  to  the  dia- 
gram the  dotted  line  CK,  showing  the  vapor  pressure  of  the  pure 
solute  at  different  temperatures.  Ice,  water  vapor  and  the  vapor  of 
the  solute  can  exist  in  the  closed  field,  COBHC  ;  the  field  for  salt 
and  vapor  can  not  extend  below  CK  ;  the  field  for  ice  and  solution 
is  now  bounded  by  D,BHOD,  the  other  fields  not  being  changed 
necessarily.  While  the  direction  and  position  of  the  boundary 
curves  may  change  a  good  deal  with  a  volatile  solid  solute  as  one 
component  there  is  but  one  displacement  that  calls  for  particular 
comment.  The  boundary  curve  OA  may  intersect  the  curve  BB, 
and  if  this  takes  place  at  a  pressure  of  less  than  one  atmosphere  we 
shall  have  the  phenomenon  of  a  solution  saturated  in  respect  to  a 


Braun's  formula,  there  appear  the  heat  of  solution  and  the  change  of  solu- 
bility with  the  temperature.  As  these  two  always  have  the  same  sign,  it  dis- 
appears from  the  equation.  This  seems  to  have  been  overlooked  both  by 
Braun  and  by  Ostwald.  Cf.  Lehrbuch  I,  1046. 

1  Favre  and  Valson,  Comptes  rendus,  79,  936,  1068  (1874);  Cf.  Mac- 
Gregor,  Zeit.  phys.  Chem.  9,  231,  236  (1892). 

'2  Comptes  rendus,  I2O,  991  (1895). 


Tico  Components  53 

solid  boiling  at  a  lower  temperature  than  the  pure  solvent.  I  know 
of  no  case  where  this  has  been  observed  but  this  proves  nothing. 
Most  solids  are  very  slightly  volatile  at  ordinary  temperatures  ;  with 
the  consequence  that  in  a  boiling  saturated  solution  the  decrease  in 
the  partial  pressure  of  the  solvent  is  not  compensated  by  the  partial 
pressure  of  the  solute.  The  conditions  necessary  for  an  example  of 
this  type  are  a  solid  with  a  high  vapor  pressure  and  a  liquid  in  which 
the  solute  shall  be  sparingly  soluble  at  the  boiling  point. 

Returning  to  the  particular  case  of  potassium  chloride  and 
water  and  the  assumption  of  a  non- volatile  solute,  it  is  clear  from 
the  diagram  that  if  a  system  composed  of  salt  and  vapor  is  sub- 
jected to  an  external  pressure  continuously  greater  than  its  own  and 
is  at  the  same  time  kept  at  constant  temperature,  there  are  three 
cases  to  be  considered  ;  when  the  temperature  is  above  the  freezing 
point  of  the  pure  solvent,  when  it  is  between  the  fusion  point  of  the 
pure  solvent  and  the  cryohydric  temperature  of  the  mixture,  and 
when  it  is  below  this  last  temperature.  In  the  first  case,  there  will 
be  compression  of  the  vapor  phase  without  condensation  until  the 
vapor  pressure  of  the  saturated  solution  is  reached  when  there  will 
be  formed  the  monovariant  system,  salt,  solution  and  vapor.  The 
pressure  will  then  remain  constant  until  either  the  salt  or  the  vapor 
disappears,  depending  on  the  relative  quantities  of  each,  leaving  the 
di variant  system,  solution  and  vapor,  or  the  one,  salt  and  solution. 
If  the  former,  further  increase  of  pressure  will  result  in  the  forma- 
tion of  the  trivariant  system,  solution.  This  will  also  be  formed 
eventually  by  the  compression  of  salt  and  solution  if  the  salt  dis- 
solves with  contraction  ;  otherwise  the  final  state  will  be  salt  and 
solution  unless  the  salt  should  be  liquefied  by  the  extreme  pressure. 
This  last  is  not  probable  as  it  could  occur  only  in  case  the  salt  ex- 
panded on  melting  or  that  what  has  been  called  the  solvent  dissolved 
in  the  salt  with  expansion  of  volume.  Between  the  freezing  point 
of  the  pure  solvent  and  the  cryohydric  temperature  of  the  system, 
the  result  will  be  the  same  if  the  salt  is  in  excess.  If  this  is  not  so, 
there  will  be  formation,  as  before,  of  solution  and  vapor  ;  but  with 
increasing  pressure,  at  length,  ice  will  separate  and  the  pressure  will 
then  remain  constant  until  the  vapor  phase  has  disappeared.  The 
pressure  will  increase  again,  the  ice  dissolving  until  there  is  left 


54  The  Phase  Rule 

only  solution.  If  the  system  is  subjected  to  pressure  at  temperatures 
lower  than  that  of  the  cryohydric  point,  there  will  be  formed,  as 
first  visible  change,  the  monovariant  system,  salt,  ice  and  vapor;  the 
vapor  will  condense  forming  the  divariant  system,  ice  and  salt,  which 
under  further  compression,  will  melt,  giving  salt,  ice  and  solution. 
If  the  salt  dissolves  with  contraction,  increase  of  pressure  will  cause 
first  one  and  then  the  other  of  the  solid  phases  to  disappear,  leaving 
only  the  solution  phase  at  last.  In  other  words,  in  all  cases  where 
the  solvent  contracts  in  fusing  and  there  is  decrease  of  volume  and 
absorption  of  heat  when  the  solid  solute  dissolves  in  the  liquid  sol- 
vent, the  final  result  of  compression  at  constant  temperature  is  the 
formation  of  the  trivariant  system,  solution.  If  these  conditions 
are  not  fulfilled,  the  course  of  events  will  be  somewhat  different  ; 
but  our  knowledge  of  the  subject  is  too  limited  to  permit  of  making 
a  detailed  consideration  of  all  possible  cases.  If  we  make  the  rather 
plausible  assumption  l  that  at  sufficiently  high  pressures  the  change 
of  the  boundary  curve  for  solid  solvent,  solid  solute,  and  solution, 
wtih  the  temperature  has  the  same  sign  as  the  corresponding  value 
for  the  boundary  curve  for  solid  and  liquid  solvent,  we  can  say  that 
the  final  result  of  compression  at  constant  temperature  will  be  form- 
ation of  solution  if  the  solvent  contracts  on  fusing  and  the  formation 
of  the  two  solid  phases  if  the  solvent  expands  when  passing  from  the 
solid  to  the  liquid  state. 

If  heat  be  withdrawn  from  the  system,  solution  and  vapor,  kept 
at  constant  volume,  the  ordinary  case  will  be  a  decrease  of  pressure 
and  temperature  until  the  boundary  curve  for  salt,  solution  and 
vapor  is  reached.  The  changes  of  pressure  and  temperature  will  be 
represented  by  that  curve  and  the  curve  for  the  two  solid  phases  and 
vapor.  The  assumptions  made  here  are  that  the  vapor  phase  does 
not  disappear  at  the  cryohydric  point  either  on  supplying  or  with- 
drawing heat  and  that  the  solute  is  more  soluble  at  high  than  at  low 
temperatures.  The  first  assumption  need  not  be  considered  as  the 
remarks  about  the  behavior  of  a  system  of  one  component  under 
similar  circumstances  apply  here  with  the  addition  of  the  solid  solute 

1  Cf.  however  the  behavior  of  sodium  chloride  and  water  under  pressure. 
Braun,  Wied.  Ann.  30,  262  (18^7). 


Two  Components  55 

as  an  extra  phase.  If  the  solute  disolves  with  evolution  of  heat,  the 
monovariant  system  formed  on  cooling  at  constant  volume  will  be 
the  one  composed  of  solid  solvent,  solution  and  vapor.  That  curve 
will  represent  the  pressures  and  temperatures  of  the  system  until  the 
cryohydric  point  is  reached.  The  change  on  codling  further  will 
be  the  same  as  in  the  previous  case. 


CHAPTER  V 

HYDRATED  SALTS 

If  we  drop  the  condition  that  the  solvent  and  solute  shall  not 
crystallize  together,  a  number  of  new  solid  phases  become  possible 
introducing  distinct  changes  in  the  conditions  of  equilibrium.  It  will 
be  best  to  confine  the  discussion  for  the  present  to  the  cases  in  which 
the  two  components  crystallize  in  definite  and  multiple  proportions, 
in  other  words  with  formation  of  compounds  and  not  of  solid  solu- 
tions. When  the  solvent  is  water,  the  compound  formed  is  called  a 
hydrate  or  a  hydrated  salt  and  the  water  is  termed  water  of  crystal- 
lization. It  is  possible  also  to  have  benzene,  alcohol,  ammonia,  hydro- 
chloric acid  and  many  other  volatile  substances  crystallizing  with  a 
practically  non-volatile  body  in  definite,  discontinuous  amounts. 
These  crystals  are  often  regarded  as  so-called  ' '  molecular ' '  com- 
pounds in  contradistinction  to  so-called  ' '  chemical ' '  compounds  ; 
but  this  distinction  cannot  be  considered  valid  until  accurate  defini- 
tions of  these  hypothetical  classes  are  given.  While  it  is  by  no 
means  certain  or  even  probable  that  a  radical  distinction  cannot  be 
drawn  between  a  hydrated  salt  and  calcium  carbonate,  for  instance, 
none  such  has  been  drawn  as  yet.  As  an  example  of  the  changes  in 
equilibrium  introduced  by  the  possibility  of  the  solvent  and  solute 
crystallizing  together  we  will  consider  the  case  of  sodium  sulfate 
which  crystallizes  from  aqueous  solutions  in  the  form  of  Na2SO4,  of 
Na2SO4  yH2O  and  of  Na2SO4  10  H,O,  depending  on  the  conditions 
of  the  experiment.  The  pressure-temperature  diagram  for  sodium 
sulfate  and  water  is  shown  in  Fig.  6.  It  is  not  drawn  to  scale. 

OOj,  OB,  OC  and  OD  are  the  curves  representing  the  equilibria 
for  hydrate,  solution  and  vapor  ;  ice,  solution  and  vapor  ;  hydrate, 
ice  and  vapor  ;  hydrate,  ice  and  solution.  O  is  the  cryohydric  point 
just  as  in  Fig.  4  and  the  remarks  in  regard  to  the  equilibrium  be- 
tween potassium  chloride  and  water  apply  to  these  four  curves  also, 
there  being  no  change  introduced  by  the  presence  of  a  hydrated  salt 


Tico  Components 


instead  of  an  anhydrous  one.  If,  however,  \ve  add  heat  to  the 
system  hydrate,  solution  and  vapor,  kept  at  constant  volume, 
there  will  be  a  rise  of  temperature  and  pressure,  the  system 
passing  along  the  curve  OOt.  At  Ot  a  new  solid  phase  ap- 
pears in  the  form  of  the  anhydrous  salt  and  there  is  present  a  new 
non variant  system  composed  of  hydrate,  anhydrous  salt,  solution 
and  vapor.  The  point  Ot  is  therefore  an  inversion  point  and  must 
be  situated  at  the  intersection  of  four  boundary  curves.  This  is  the 
case  and  the  curves  OOP  AO,,  COj,  D,O,,  represent  the  possible 
temperatures  and  pressures  for  the  four  monovariant  systems, 
hydrate,  solution  and  vapor ;  anhydrous  salt,  solution  and  vapor  ; 
hydrate,  anhydrous  salt  and  vapor  ;  hydrate,  anhydrous  salt  and 
solution. 

P 


FIG.  6. 

The  temperature  at  which  the  nonvariant  system  can  exist  is 
32.6°  and  the  pressure  30.8  millimeters  of  mercury.1  Any  con- 
tinued supply  or  withdrawal  of  heat  or  work  results  finally  in  the 


Cohen,  Zeit.  phys.  Chem.  14,  90  ( 1894). 


58  The  Phase  Rule 

disappearance  of  one  of  the  four  phases  without  change  of  temper- 
ature or  pressure.  The  curve  AOP  if  prolonged,  will  be  found  to 
lie  below  the  curve  OO^  Since  it  represents  a  labile  equilibrium  at 
all  temperatures  below  32.6°,  we  see  that  here  the  less  stable  system 
has  a  lower  vapor  pressure  than  the  more  stable  one.  Although 
this  is  contrary  to  our  previous  experiences,  it  is  a  perfectly  general 
result.  The  more  stable  system  is  the  one  with  the  lesser  concen- 
tration and  therefore  the  greater  vapor  pressure.  There  are  two 
forces  acting  in  opposite  directions,  the  tendency  of  the  vapor  to 
distill  from  a  place  of  higher  to  one  of  lower  pressure,  and  the 
tendency  of  the  excess  of  solute  to  precipitate  from  the  more  con- 
centrated solution.  Either  of  these  changes  would  bring  about 
equilibrium  alone  but  the  second  is  the  stronger.  If  the  two  solu- 
tions are  placed  under  a  bell- jar  we  shall  get  distillation  and  the 
more  concentrated  solution  as  the  stable  one  ;  if  the  two  solid  phases 
are  brought  into  the  same  solution  the  stable  form  is  the  one  which 
is  in  equilibrium  with  the  more  dilute  solution.  The  important 
factor  in  determining  the  equilibrium  is  the  change  in  the  vapor 
pressure  of  the  solute.  The  more  concentrated  the  solution  the 
higher  the  partial  pressure  of  the  solute  and  stable  equilibrium  is 
reached,  experimentally,  when  the  more  stable  solid  phase  is  pres- 
ent, the  one  with  the  lower  vapor  pressure.  When  the  two  satura- 
ted solutions  are  connected  by  the  vapor  phase  only,  there  is  distilla- 
tion of  water  in  one  direction  and  of  solute  in  the  other  ;  but  the 
velocity  of  the  first  reaction  being  much  greater  than  that  of  the 
second  the  two  solutions  come  to  the  same  concentration  before  any 
measurable  amount  of  the  solute  has  been  transferred.  It  is  to  be 
noticed  that  although  the  partial  pressure  of  the  solute  may  be  in- 
finitely small,  it  exerts  the  controlling  influence  in  regard  to  the 
stability  of  the  system  when  both  solid  phases  are  added.  '  It  is  too 
often  assumed  that  a  value  may  be  neglected  in  respect  to  all  meas- 
urements because  it  can  be  neglected  in  respect  to  one. 

From  the  existence  of  these  two  solubility  curves  at  a  given 
temperature,  say  30° ,  it  is  clear  that  it  is  not  sufficient  to  speak  of  a  satu- 
rated solution  without  defining  the  solid  phase  with  respect  to  which 
it  is  saturated.1  As  has  been  said,  the  solution  saturated  at  30°  with 


1  Ostwald,  L,ehrbuch  I,  1036. 


7Y.v  Components  59 

respect  to  anhydrous  sodium  sulfate  contains  more  of  the  solute  than 
the  solution  saturated  with  respect  to  Na2SO4ioH2O  and  is  therefore 
in  a  state  of  labile  equilibrium,  stable  only  so  long  as  no  solid 
hydrated  salt  is  present.  There  is  yet  another  saturated  solu- 
tion which  can  exist  at  this  temperature,  the  solid  phase  being 
Na-jSOjH.,0.  This  system  is  instable  both  with  respect  to  the 
anhydrous  salt  and  to  the  decahydrate,  being  more  soluble  than  either. 
The  vapor  pressure  of  the  solution  is  therefore  less  than  that  of 
either  of  the  other  solutions  and  is  represented  by  the  dotted  line 
KKr  This  monovariant  system  differs  from  the  other  two  in  that 
at  no  temperature  does  it  represent  a  state  of  stable  equilibrium.  In 
this  it  is  analogous  to  the  yellow  phosphorus  which  is  labile  both  as 
solid  and  as  liquid.  The  crystals  of  Na2SOjH2O  can  be  obtained 
by  addition  of  alcohol  to  a  mixture  of  sodium  sulfate  and  water.1 

Starting  from  the  nonvariant  system  it  is  possible  to  reach  the 
monovariant  system,  hydrate,  anhydrous  salt  and  vapor,  by  keeping 
the  external  pressure  constantly  less  than  that  of  the  nonvariant 
system  until  the  whole  of  the  solution  has  disappeared.  According 
to  the  Phase  Rule  this  new  system  must  have  a  definite,  unchanging 
vapor  pressure  at  each  temperature  so  long  as  the  three  phases  are 
present,  entirely  independent  of  the  absolute  mass  of  any  of  them. 
This  is  found  to  be  the  case  experimentally.  If  the  external  pres- 
sure upon  the  system  be  increased  there  will  be  condensation  of 
water  vapor  and  formation  of  hydrate  at  the  expense  of  the  anhydrous 
salt.  If  it  be  decreased  the  hydrate  will  lose  water,  efflorescing  as 
it  is  called.  While  this  increase  and  decrease  of  external  pressure 
can  be  brought  about  theoretically  by  considering  the  system  in  a 
closed  vessel  with  a  movable  piston,  such  as  a  barometer  tube,  and 
increasing  or  decreasing  the  volume  of  the  vapor  phase  and  therefore 
of  the  system,  it  is  often  convenient,  practically,  to  work  at  constant 
volume.  This  can  be  done  by  introducing  into  the  vapor  phase 
some  substance,  such  as  pure  water  which  will  give  off  water  vapor 
at  a  higher  pressure  than  that  of  the  system  under  consideration  or, 
in  the  reverse  case,  by  introducing  some  such  substance  as  strong 
sulfuric  acid  which  will  take  up  water  at  a  less  pressure  than  that  of 
the  system.  Another  way  in  which  the  external  pressure  may  be 


Dammer,  Handbuch  II  2,  156. 


60  The  Phase  Rule 

kept  less  than  the  equilibrium  pressure  of  the  system  without  using  a 
closed  vessel  and  a  movable  piston  is  by  passing  a  current  of  dried 
air  or  other  gas  through  the  vessel  in  which  the  system  is  placed. 
By  this  means  the  water  vapor  is  carried  off  as  fast  as  formed  and 
there  will  be  no  chance  for  the  concentration  of  water  in  the  vapor 
phase  to  rise  to  the  value  corresponding  to  the  equilibrium  pressure 
of  the  system.  This  last  method  is  of  use  only  as  a  means  of  re- 
moving a  given  phase  rapidly  and  cannot  be  employed  for  an  accur- 
ate study  of  equilibrium  because  it  introduces  new  components  into 
the  system,  thereby  changing  the  whole  problem. 

The  vapor  pressures  of  the  system,  hydrate,  anhydrous  salt  and 
vapor,  cannot  be  higher  than  those  for  the  solution  saturated  in 
respect  to  the  hydrate,  because  water  would  then  condense,  building 
the  latter  system  at  the  expense  of  the  former,  which  does  not  hap- 
pen experimentally.  The  curve  OjC  can  not,  therefore,  lie  above 
the  curve  C^O,  and  often  lies  a  good  way  below  it.  In  Table  IX 

TABLE  IX 


Temp.   20° 

Solution  i 

Salt 

Temp.  20° 

Solution 

1 

Salt 

CaCl26H2O 
SrCl26H2O 
MnCl24H2O 
NiCl26H2O 
CoCl26H2O 
NaBr4H2O 
SrBr.,6H2O 

5-4 
11.4 
8.0 
8.0 
9.0 
9-6 
9-1 

2-3 

5-6 
3-8 
4-6 
4.0 
7.6 
i-7 

Na2CO3i  2H2O 
Na2SO4ioH2O 
Na2SOjH2O 
MgSO47H2O 
CuSO45H2O* 
MgCl26H2O 
NaI4H90 

16.0 

15-7 
15.0 

14-5 
58.0 

5-7 
5-4 

10.  I 

13-9 
10.5 
10.3 
30.0 
1.8 
i-5 

*  Vapor  pressure  at  45°  instead  of  20°. 

are  the  vapor  pressures  in  millimeters  of  mercury  of  certain  satura- 
ted solutions  and  of  the  partly  effloresced  crystals.  As  will  be  no- 
ticed the  differences  between  the  two  columns  are  quite  considerable 
in  many  instances.  In  some  cases  the  two  sets  of  values  coincide 
within  the  limit  of  experimental  error,  as  is  shown  in  Table  X  l  and 
still  more  strikingly  in  Table  XI.2 


'Lescceur,  Ann.  chim.  pliys.  (6)  19,  533  ;  21,  511  (1890);  (7)  2,  78  (1894). 
2  Joannis,  Comptes  reudus,  XIO,  238  (1890). 


Tsro  Components 
TABLE  X 


Solution       Salt        Temp. 


MgBr/iH/) 
MgBr~6H,O 


BaIj6H,O 


10.7 
124.0 

3-4 

166.0 

10.0 

I22.O 

8.4 

58-0 

5-o 


10.6 
124.0 

3-3 
166.0 

124.0 

8.4 

60.0 

5-o 


202. 0  200.0 


20° 
60 
2O 

IOO 
20 
60 
2O 
60 
2O 

IOO 


TABLE  XI 


igXH,Xa+o.46ogXHs 
0.97  igXH,Xa+o.o29gXa 

o.  io8gXHsXa-f-o.892gXa 
o.o43gXH3Xa+o.957gXa 


o.  7gXHjXa +o.  30X3 
o.39gXH,Xa+o.6igXa 
o.  i9gNH,Xa+o  SigXa 


Pressure        Temp. 


169.7 

169.7 

169.7 

169.7 

169.7 

169-65 

117.0 

"7-3 
117.0 
1 17. 1 


O^ 
O 

o 
o 
o 
o 

—  IO 

—  : 

— 10 

—  IO 


The  data  in  Table  XI  are  very  interesting  because  they  were 
collected  for  the  express  purpose  of  showing  that  the  vapor  pressures 
of  an  efflorescing  compound  could  have  the  same  value  as  the  vapor 
pressures  of  the  corresponding  saturated  solution  for  a  series  of  tem- 
peratures. Roozeboom1  has  stated  that,  in  his  opinion,  this  could 
occur  only  at  the  inversion  temperature,  but  the  facts  do  not  seem  to 
have  borne  him  out  in  this  view.  It  is  clear  that  these  two  curves 
can  not  actually  coincide ;  but  the  difference  between  them  may  be  a 
difference  in  the  partial  pressures  and  not  in  the  total  vapor  pressure. 
In  a  case  of  this  sort  the  concentration  in  the  vapor  phase  of  the  so- 


1  Comptes  rendos.  IIO,  135  (1890). 


62  77?^  Phase  Rule 

called  non-volatile  component  is  of  great  importance  in  spite  of  its 
not  having  a  measurable  value.  All  that  the  Phase  Rule  states  is 
that  the  vapor  phase  for  the  one  monovariant  system  can  not  be 
identical  in  every  respect  with  the  vapor  phase  for  the  other  mono- 
variant  system.  When  the  curves  OiO  and  OjC  coincide  within  the 
limits  of  experimental  error,  it  seems  probable  that  the  continuation 
of  AO  will  lie  below  O,C.  Under  these  circumstances  if  a  mixture 
of  anhydrous  and  hydrated  salt  be  placed  in  a  beaker  and  the  insta- 
ble  saturated  solution  of  the  anhydrous  salt  in  another  beaker,  both 
under  the  same  bell-jar,  there  will  be  distillation  from  the  first  to  the 
second  and  spontaneous  formation  of  an  instable  system  at  the  ex- 
pense of  the  stable  one.  As  no  instance  of  this  has  yet  been  studied 
it  is  hardly  worth  while  to  draw  any  more  conclusions  till  this  has 
been  done. 

Little  is  known  about  the  vapor  pressures  of  the  system,  sodium 
sulfate  heptahydrate,  anhydrous  sodium  sulfate  and  vapor,  though 
this  curve  of  course  lies  below  KKj.1  This  curve  is  not  given  in 
the  diagram  but  it  exists  below  25°,  the  point  at  which  the  continu- 
ation of  AO,  would  cut  KKr  Above  this  temperature  we  should  ex- 
pect that  the  crystals  of  sodium  sulfate  heptahydrate  would  change 
into  anhydrous  salt,  solution  and  vapor,  instead  of  efflorescing.  It 
is  not  known  whether  an  equilibrium  can  be  established  between  the 
two  hydrated  sodium  sulfates  and  vapor,  but  this  could  probably  be 
realized.  Usually  the  decahydrate  effloresces  to  the  anhydrous  salt 
without  formation  of  the  heptahydrate.  Although  this  question  of 
efflorescence  has  not  been  worked  out  very  carefully  there  is  little 
doubt  that  a  hydrated  salt  effloresces  normally  with  formation  of  the 
solid  phase  which  appears  at  the  next  higher  stable  inversion  point. 
Since  the  vapor  pressure  of  the  system  hydrate,  effloresced  salt 
and  vapor,  increases  experimentally  with  rising  temperature,  it  fol- 
lows from  the  Theorem  of  L,e  Chatelier  that  the  decomposition  of 
the  hydrate  into  its  dissociation  products  is  accompanied  by  an  ab- 
sorption of  heat  or  that  the  formation  of  the  hydrate  is  attended  by 
an  evolution  of  heat.  This  can  be  shown  experimentally  by  adding 
an  excess  of  anhydrous  sodium  carbonate  to  water  when  a  distinct 
rise  of  temperature  will  be  observed.  If  water  be  added  to  crystal- 


1  Cf.  Table  IX. 


Two  Components  63 

lized  sodium  carbonate,  there  will  be  a  fall  of  temperature  because 
the  solubility  of  the  hydrate  increases  with  rising  temperature  and 
therefore  heat  is  absorbed  in  the  process  of  dissolving.  If  the  heat 
evolved  in  the  change  from  the  anhydrous  to  the  hydrated  salt  be 
greater  than  the  heat  absorbed  in  passing  from  the  hydrate  to  the 
saturated  solution,  there  may  be  heat  evolved  in  the  change  from 
anhydrous  salt  to  saturated  solution  of  the  h>-drate  even  though  the 
latter  be  more  soluble  in  warm  than  in  cold  water.  This  is  probably 
what  Ostwald  has  in  mind  when  he  saj-s  :l  "  It  is  a  familiar  rule  that 
salts  which  dissolve  in  water  with  evolution  of  heat  cr ystallize  with 
water  of  crystallization  ;  anhydrous  salts  dissolve  with  absorption  of 
heat."  In  this  form  the  statement  is  incorrect.*  It  is  also  indefinite 
because  Ostwald  has  not  specified  whether  he  refers  to  the  therm ody- 
namical  or  the  thermochemical  heat  of  solution. 

The  curve  OaD,  representing  the  equilibrium  between  hydrate, 
anhydrous  salt  and  solution,  introduces  no  new  features  and  needs 
no  discussion.  It  will  be  noticed  that  in  the  diagram,  Fig.  6,  the 
curve  KKlt  for  the  system,  Xa2SO47H,O,  solution  and  vapor, 
cuts  at  some  unknown  point  the  curve  O,C,  for  the  system, 
XajSOjoHjO,  NajSO4  and  vapor.  This  intersection  does  not  rep- 
resent a  new  inversion  point  because  it  is  not  the  locus  of  the  curves 
for  four  inonovariant  systems  but  the  intersection  of  two  curves  for 
two  monovariant  systems  which  have  only  one  phase  in  common, 
the  vapor  phase.  If  we  could  measure  the  vapor  pressure  of  the 
sodium  sulfate  we  should  find  that  at  the  intersecting  point  the  total 
vapor  pressures  of  the  two  s}*stems  were  equal  but  that  the  partial 
pressures  were  not.  There  is  only  an  apparent  identity  in  the  two 
vapor  phases  and  therefore  no  reason  for  assuming  that  there  might 
be  equilibrium  between  the  two  systems 

In  order  to  determine  the  boundaries  of  the  fields  in  which  the 
di variant  systems  can  exist,  I  have  added  the  dotted  line  BB^  which 
is  the  pressure-temperature  curve  for  pure  water  in  equilibrium  with 
its  own  vapor.  Solution  and  vapor  can  be  in  stable  equilibrium  in 
the  field  AO,OBBj ;  solution  and  anhydrous  salt  in  AO,D, ;  solution 


1  Lehrbuch  II,  800. 

1  Cf.  Roozeboom,  Recaei)  Trav.  Pays-Bas,  8,  in  (1889). 


64  The  Phase  Rule 

and  hydrate  in  D2OtOD  ;  anhydrous  and  hydrated  salt  in 
hydrated  salt  and  vapor  in  C^OCOj ;  while  anhydrous  salt  and  vapor 
can  exist  in  the  field  bounded  by  AOjC  and  the  temperature  axis. 
The  distribution  of  the  fields  round  the  point  O  is  practically  the 
same  as  for  the  system,  potassium  chloride  and  water,  substituting 
Na2SO4ioH2O  for  KC1.  The  only  difference  is  that  the  di variant 
system,  hydrate  and  vapor,  can  not  exist  at  pressures  lower  than 
those  of  the  curve  O,C.  Since  we  are  considering  stable  equilibrium 
only,  the  hydrate  Na.2SO47H2O  does  not  enter  into  the  discussion. 

From  the  diagram  we  can  predict  the  behavior  of  hydrates  of  this 
type  when  heated.  When  the  temperature  of  the  point  Oj  is 
reached  the  hydrate  will  seem  to  melt  with  precipitation  of  salt, 
forming  the  nonvariant  system,  hydrate,  anhydrous  salt,  solution 
and  vapor,  a  further  addition  of  heat  causing  the  disappearance  of 
the  hydrate.  The  temperature  at  which  this  change  takes  place  in 
such  salts  as  sodium  sulfate  and  sodium  carbonate  is  not  the  melting 
point  of  the  hydrate  but  the  inversion  point.  Later  we  shall  study 
hydrates  which  have  true  melting  points.  Hydrate  and  vapor,  be- 
ing a  divariant  system,  can  exist  at  more  than  one  pressure  for  a 
given  temperature,  and  it  may  be  well  to  specify  in  words  the  condi- 
tions under  which  hydrates  change  when  exposed  to  the  air.  We 
have  already  seen  that  an  anhydrous  salt  is  permanent  when  the 
pressure  of  water  vapor  in  the  atmosphere  is  less  than  the  pressure 
of  the  saturated  solution  and  is  deliquescent  when  it  exceeds  that 
value.  Since  the  field  for' hydrate  and  vapor  lies  between  O,C  and 
OiOC  it  follows  that  at  all  temperatures,  above  that  of  the  cryohydric 
point  O,  a  hydrated  salt  will  effloresce  when  the  partial  pressure  of 
water  vapor  in  the  atmosphere  falls  below  the  pressure  of  the  sys- 
tem, hydrate,  effloresced  salt  and  vapor  ;  will  deliquesce  when  it  ex- 
ceeds the  vapor  pressure  of  the  saturated  solution  and  will  be  per- 
manent at  intermediate  values.  Whether  the  range  of  pressures 
over  which  the  hydrate  is  permanent  is  an  extended  one  or  not  de- 
pends on  the  relative  positions  of  the  curves  OjO  and  O,C  at  the 
temperature  of  the  experiment.  At  temperatures  below  that  of  the 
cryohydric  point  the  hydrate  will  effloresce  under  the  same  con- 
ditions as  before  ;  but  if  the  pressure  of  aqueous  vapor  in  the  atmos- 
phere exceeds  the  value  of  OC  for  that  temperature,  there  will  be  a 
precipitation  of  ice,  the  hydrate  remaining  unchanged. 


Two  Components  65 

The  general  results  which  have  just  been  obtained  are  not  con- 
fined to  reactions  between  salts  and  water.  The  same  phenomena 
will  recur  in  all  cases  in  which  the  solute  and  solvent  form  a  solid 
compound,  provided  that  it  is  not  possible  to  have  two  co-existing 
liquid  phases,  a  case  which  will  be  treated  by  itself.  It  is  to  be  re- 
membered that  if  each  component  has  a  perceptible  vapor  pressure 
this  will  have  an  effect  on  the  extent  of  the  fields  in  which  the  dif- 
ferent divariant  systems  can  exist.  The  phenomenon  of  a  constant 
vapor  pressure  at  each  temperature  for  the  system,  hydrate,  efflor- 
esced salt  and  vapor,  will  occur  in  all  cases  in  which  a  solid  com- 
pound dissociates  into  a  solid  and  a  vapor.  This  has  been  shown  to 
hold  for  the  compounds  formed  by  the  action  of  ammonia  on  the 
silver  haloids,1  on  ammonium  bromide2  and  on  metallic  sodium.3  A 
more  interesting  case  than  any  of  these  is  the  dissociation  of  calcium 
carbonate  into  calcium  oxide  and  carbonic  acid.  There  is  equilib- 
rium between  the  three  phases  only  at  one  pressure  for  each  temper- 
ature, at  the  dissociation  pressure  so-called.4  The  divariant  system, 
calcium  carbonate  and  carbonic  acid,  on  the  other  hand,  can  exist  at 
a  given  temperature*  under  any  pressure  between  the  dissociation 
pressure  and  the  pressure  at  which  carbonic  acid  condenses  to  a 
liquid  in  the  presence  of  calcium  carbonate.  Isambert5  found  that 
at  temperatures  in  the  neighborhood .  of  1100°  C,  the  dissociation 
pressure  of  barium  carbonate  was  so  low  that  the  salt  was  prac- 
tically not  decomposable  by  heat  in  open  vessels  owing  to  the  amount 
of  carbonic  acid  in  the  air.  By  passing  a  current  of  nitrogen 
through  the  apparatus,  thus  keeping  the  partial  pressure  of  the  car- 
bonic acid  down  towards  zero,  the  compound  effloresced  completely 
with  formation  of  barium  oxide.  The  same  effect  was  obtained  by 
mixing  powdered  carbon  with  the  salt.  The  carbon  combined  with 
the  carbonic  acid  forming  carbon  monoxide  and  keeping  the  partial 
pressure  of  the  carbonic  acid  at  a  minimum  value.  It  may  be  urged 
that  all  this  is  not  strictly  analogous  to  the  behavior  of  a  hydrated 


1  Isambert,  Comptes  rendus,  66,  1259  (1868)  ;  70,  456  (1870). 

2  Roozeboom,  Recueil  Trav.  Pays-Bas,  4,  355  (1885). 
3Joanuis,  Comptes  rendus,  IXO,  238  (1890). 

4Debray,  Ibid.,  64,  605  (1867)  ;  Le  Chatelier,  Ibid.  IO2,  1243  (1886). 
5  Ibid.,  86,  332(1878). 
5 


66 


The  Phase  Rule 


salt  because  the  dissociation  pressure  of  calcium  carbonate  is  prac- 
tically zero  at  room  temperatures,  and  at  higher  temperatures  car- 
bonic acid  is  a  gas  and  not  a  vapor  ;  but  it  has  not  been  shown  that 
this  fact  introduces  any  fundamental  distinction. 

Since  concentrations  are  more  easily  measured  than  vapor  pres- 
sures, it  is  sometimes  advantageous  to  use  the  concentration  and 
temperature  as  co-ordinates.  The  resulting  diagram  is  not  so  gen- 
eral as  the  pressure-temperature  diagram  because  it  can  be  applied 
only  to  phases  of  varying  concentration  ;  but  it  conveys  certain  in- 
formation that  is  only  obtainable  indirectly  from  the  other  diagram. 
Since  people  have  studied,  almost  exclusively,  monovariant  systems 
composed  of  a  practically  non- volatile  solute  and  a  volatile  solvent, 


FIG.  7. 

the  concentration  of  the  liquid  phase  is  the  only  one  that  is  ordinar- 
ily tabulated.  This  is  not  necessary  and  is  due  to  our  very  complete 
ignorance  of  the  composition  of  vapor  phases  containing  two  com- 
ponents. In  Fig.  7  is  shown  the  concentration-temperature  diagram 
for  sodium  sulfate  and  water.  The  ordinates  are  grams  of  sodium 
sulfate  in  one  hundred  grams  of  water  and  the  drawing  is  to  scale  ; 
but  the  distance  equal  to  one  degree  is  ten  times  as  great  for  the 
region — 1°  too0  as  for  the  rest  of  the  diagram.  The  system  is 
supposed  to  be  under  its  own  pressure. 

The  curve  BO  is  the  fusion  curve  showing  the  concentrations  and 
temperatures  at  which  the  solution  can  be  in  equilibrium  with  ice. 
This  curve  runs  to  the  left  because  increasing  concentration  causes 
increasing  depression  of  the  freezing  point.  At  the  cryohydric 


Two  Components  67 

point  O,  the  hydrate  Xa3SO4ioH,O  begins  to  crystallize  out  and  we 
have  the  first  inversion  point,  the  temperature  being  — 0.7°  and  the 
concentration  being  about  4.8  grams  per  hundred  of  water.  With 
increasing  concentration  the  system  passes  along  the  curve  OO,,  the 
solubility  curve  for  NajSOjoHjO.  At  O,  anhydrous  sodium  sulfate 
crystallizes  from  the  solution  forming  the  second  non variant  system, 
the  temperature  being  32.6°  and  the  concentration  about  49. 8  grams. 
Beyond  this  temperature  the  concentration  does  not  increase  because 
anhydrous  sodium  sulfate  dissolves  with  evolution  of  heat.  The 
curve  O,A  is  the  concentration-temperature  curve  for  the  solution 
saturated  in  respect  to  anhydrous  sodium  sulfate.  The  line  OO,A  is 
therefore  not  a  continuous  curve  but  two  curves  meeting  at  an  angle. 
This  is  a  characteristic  phenomenon  and  one  may  say  that  in  all  cases 
where  a  solubility  curve  shows  a  "  break  "  or  discontinuous  change 
of  direction  some  new  substance  is  separating  from  the  solution. 
Conversely,  if  the  solubility  curve  has  a  continuous  change  of  direc- 
tion the  solution  is  saturated  in  regard  to  the  same  substance  ;  but 
this  does  not  apply  to  the  intersection  of  a  fusion  and  solubility 
curve,  a  case  which  will  receive  special  consideration.  The  curve 
OO,  has  been  realized  only  with  great  difficulty  bej-ond  the  inversion 
point ;  but  the  curve  AO,  has  been  followed  some  fifteen  degrees  be- 
low 32.6°.  The  solution,  thus  obtained,  contains  more  of  the  solute 
than  the  solution  saturated  at  the  same  temperature  in  respect 
to  NajSOjoH^O  and  is  therefore  instable  in  the  presence  of  a  crys- 
tal of  the  hydrate.  The  curve  KK,  shows  the  concentrations  at 
different  temperatures  of  solutions  saturated  in  respect  to 
Na^OjHjO.  As  this  curve  lies  above  the  broken  line  OO,A  the 
equilibrium  is  always  labile.  At  the  intersection  of  AO,  with  KK, 
in  the  neighborhood  of  25°  there  is  possible  the  labile  nonvariant 
system,  Na^O4,  Na,SO47H,O  solution  and  vapor,  instable  in  respect 
to  XaJSO4ioHaO.  I  am  not  aware  that  this  has  been  observed  ex- 
perimentally. 

In  the  field  bounded  by  AO,OB  and  the  temperature  axis  there 
is  stable  equilibrium  between  unsaturated  solution  and  vapor.  If  a 
solution  having  the  concentration  and  temperature  represented  by 
the  point  M  be  cooled  in  a  closed  vessel,  the  changing  state  of  'the 
system  will  be  represented  by  the  horizontal  dotted  line  MM,.  At 


68  The  Phase  Rule 

M,  the  solution  has  the  concentration  of  the  saturated  solution  and 
on  further  cooling  the  solid  phase  should  appear,  the  system  passing 
then  along  the  line  MjO.  If  the  solution  be  not  jarred,  it  often  hap- 
pens that  no  precipitate  is  formed  and  the  state  of  the  system  is  rep- 
resented by  some  point  on  the  prolongation  of  MM,.  As  the  solu- 
tion contains  more  dissolved  substance  than  the  saturated  solution  it 
is  instable  with  respect  to  the  solid  phase  and  is  said  to  be  supersat- 
urated.1 In  many  cases  shaking  will  cause  a  precipitation  of  the 
excess  ;  but  the  only  certain  method  of  causing  crystallization  is 
the  addition  of  a  crystal  of  the  substance  in  respect  to  which  the 
solution  is  supersaturated  or  a  crystal  of  an  isomorphous  compound2. 
The  phenomenon  of  supersaturation  is  not  confined  to  salts  in  water, 
being  a  more  or  less  developed  characteristic  of  all  solutions.  Sub- 
stances differ  very  greatly  in  the  ease  with  which  they  form  super- 
saturated solutions  and  very  little  is  known  of  the  cause  of  this 
behavior.  In  aqueous  solutions  it  may  be  said,  as  a  rule,  that  the 
salts  which  separate  with  water  of  crystallization  form  supersatura- 
ted solutions  more  readily  than  salts  which  separate  in  the  anhy- 
drous state.  Sodium  sulfate,  sodium  carbonate  and  the  alums  are 
good  instances  of  the  first  type,  potassium  nitrate  and  sodium 
chloride  of  the  second.  This  is  not  a  general  statement  because 
silver  nitrate  and  sodium  chlorate,  for  instance,  crystallize  in  the 
anhydrous  form  and  yet  form  supersaturated  solutions  with  great 
readiness,  to  say  nothing  of  the  fact  that  most  obstinate  cases  occur 
with  organic  solutes  in  organic  solvents.  Ostwald  *  has  pointed  out 
that  there  is  an  evident  connection  between  the  power  to  form  large 
crystals  and  the  tendency  to  form  supersaturated  solutions.  A  salt 
which  forms  highly  supersaturated  solutions  is  one  which  tends  to 
precipitate  only  on  a  crystal  already  present  while  a  salt  which  forms 
supersaturated  solutions  with  difficulty  is  one  which  has  a  great 
tendency  to  spontaneous  crystallization  in  any  and  all  parts  of  the 
solution.  The  result  in  the  latter  case  will  be  a  host  of  small  crys- 
tals ;  in  the  former,  a  single  large  one. 


1  Cf.  Ostwald,  Lehrbuch  I,  1036-1039. 

2  Gernez,  Beiblatter,  2,  241  (1878). 
;tL,ehrbuch  I,  1039. 


Two  Components  69 

The  method  of  representing  concentrations  and  temperatures 
adopted  in  Fig.  7  is  the  usual  one ;  but  it  is  not  the  best.  It  is  pos- 
sible to  represent  in  the  diagram  the  fusion  curve  of  ice  in  presence 
of  salt,  but  not  the  fusion  curve  of  salt  in  presence  of  water,  be- 
cause, at  the  melting  point  of  the  pure  salt,  the  ratio  of  salt  to  water 
becomes  infinite  and  can  not  be  shown  in  a  finite  diagram.  This 
can  best  be  done  in  the  manner  pointed  out  by  Gibbs,1  the  total  mass 
of  the  two  components,  expressed  in  any  units  whatsoever,  being 
kept  constant.  The  concentrations  can  then  be  represented  by 
points  on  a  horizontal  line  of  definite  length  and  the  temperature  on 
an  ordinate  perpendicular  to  it.  While  it  is  more  convenient  to  plot 
the  concentrations  as  so  many  grams  of  either  component  in  one 
hundred  grams  of  the  mixture  because  the  measurements  are  made 
that  way  and  because  there  are  no  assumptions  made  in  so  doing,  it 
is  found  better  in  practice  to  adopt  another  scale.  It  is  found  ex- 
perimentally that  one  gram  of  one  substance  is  not  equivalent  chem- 
ically to  one  gram  of  another  substance,  and  it  is  more  rational  to 
use  the  chemical  unit,  the  reacting  weight,  rather  than  the  weight 
unit,  the  gram.  The  concentrations,  are  then  expressed  in  reacting 
weights*  of  either  component  per  hundred  reacting  weights  of  the 
mixture.  In  this  particular  case  one  hundred  and  forty-two  grams 
of  sodium  sulfate  are  taken  as  equivalent  to  eighteen  grams  of  wa- 
ter. The  system  is  in  each  case  supposed  to  be  under  its  own  vapor 
pressure,  at  constant  volume.  Van  Rijn  van  Alkemade*  prefers  to 
treat  the  system  as  if  always  under  constant  atmospheric  pressure  ; 
but  this  does  not  seem  advisable,  as  in  that  case  we  can  not  consider 
the  intersection  of  two  curves  as  representing  a  non  variant  system. 
Since  the  determinations  are  usually  made  in  open  vessels,  the  sys- 
tem does,  as  a  matter  of  fact,  exist  under  a  constant  atmospheric 
pressure,  but  that  involves  bringing  in  the  air  as  another  component. 
Strictly  speaking  we  do  not  have  equilibrium  when  working  in  an 
open  vessel  unless  the  partial  pressure  of  each  component  in  the  at- 
mosphere is  equal  to  the  partial  pressure  of  the  same  component  in 

1  Trans.  Conn.  Acad.  3,  178(1876);  Cf.  Roozeboom,  Zeit.  phys.  Chem.  la, 
359  (»893)  ;  Konowalow,  Wied.  Ann.  14.  34  (1881);  Alexejew,  ItwL  28,  305 
[886  . 

1  Bancroft,  Phys.  Rev.  3,  25  (1895). 
3ZdL  phys.  Chem.  II,  291  (1893). 


yo 


The  Phase  Ride 


equilibrium  with  the  solution  and  then  we  are  really  working  at 
constant  volume  with  an  enormously  large  vapor  space.  As  the 
change  of  solubility  with  the  pressure  is  very  slight  there  will  be  no 
experimental  difference  between  the  composition  of  a  liquid  phase 
under  its  own  vapor  pressure  and  under  a  pressure  of  seventy-six 
centimeters  of  mercury. 


ft 


9+   A\ 


FIG.  8. 

In  Fig.  8  is  shown  the  equilibrium  between  water  and  sodium 
sulfate.  The  extreme  left  of  the  diagram  represents  one  hundred 
reacting  weights  of  water  and  zero  reacting  weights  of  sodium  sul- 
fate. The  letters  have  the  same  significance  as  in  Fig.  7,  the  curves 


BO,  OO,, 


KKj  representing  the  solutions  in  equilibrium  with 


ice,  Na2SO4ioH2O,  Na2SO4  and  Na2SOjH2O  respectively.  The 
curve  OjA  if  prolonged  must  slant  to  the  right  until  it  cuts  the  right 
hand  ordinate  at  the  fusion  temperature  of  sodium  sulfate,  about 
860°  ;  but  this  portion  of  the  curve  has  not  been  studied.  In  this 
diagram  the  field  in  which  the  unsaturated  solution  and  vapor  exist 
in  stable  equilibrium  lies  above  the  curves  BOO,A. 
1  The  letter  A  is  missing  from  Fig.  8. 


Two  Components 


In  the  case  of  sodium  sulfate  there  are  only  two  hydrates 
known,  and  one  of  these  represents  always  a  state  of  labile  equilibri- 
um. Neither  of  these  conditions  is  general  and  in  the  equilibrium 
between  calcium  chloride  and  water1  we  have  the  existence  of  many 
hydrates,  most  of  which  can  exist  in  stable  equilibrium  with  solution 
and  vapor  at  some  pressures  and  temperatures.  Fig.  9  shows  the 


FIG.  9. 

pressure-temperature  diagram  for  calcium  chloride  and  water  ;*  and 
Figs.  10  and  1 1  parts  of  the  same  diagram  on  a  larger  scale.  The 
ordinates  in  Fig.  10  are  millimeters  of  mercury;  in  Fig.  n,  centi- 
meters of  mercury.  The  possible  solid  phases  are  ice,  CaCl,6H,O, 
two  modifications  of  the  tetrahydrate  CaCl;4HJO»  and  CaCl^H^O/S, 
CaCl,2H2O,  CaCl,H2O  and  CaCl,.  The  equilibrium  between  anhy- 
drous calcium  chloride,  solution  and  vapor,  has  not  been  studied  in 
detail  owing  to  the  high  temperature  at  which  this  system  first  be- 
comes possible.  The  curves  BOHCD,  HK,  DF,  FKL  and  \M 
represent  the  pressures  and  temperatures  at  which  there  can  coexist 
vapor  and  solutions  saturated  in  regard  to  CaCl26H,O,  CaCl,4H1Oar, 
CaCl^HjO^,  CaCl,2H,O  and  CaCljH^O  respectively,  while  the  curve 
AB  for  the  vapor  pressures  of  ice  may  be  assumed  to  represent  the 
system  consisting  of  ice,  solution  and  vapor.  The  curves  IH,  JD, 
NK  and  PL  represent  the  systems  CaCl^HjO,  CaCl^HjOo'  and 
vapor  ;  CaCl^H.O,  CaCl^HjO/S  and  vapor  ;  CaCl^HjOff,  CaCl^H^O 
and  vapor  ;  and  CaCl22H,O,  CaCljHjO  and  vapor.  The  solubility 

1  Roozeboom,  Rec.  trav.  chim.  8,  i  ( 1889)  ;  ZeiL  phys.  Chem.  4,  31  (188^). 
-  The  letter  P  is  missing  from  Fig.  9. 


7- 


The  Phase  Rule 


of  the  hexahydrate  increases  so  rapidly  with  the  temperature  that 
the  vapor  pressure  of  the  saturated  solution  passes  through  a  maxium 
at  28.5°,  represented  by  the  point  O  and  decreases  from  there  to  the 
point  C,  at  30.2°,  the  fusion  point  of  the  hexahydrate.  At  higher 


1 


L 


FIG.   10. 

temperatures  than  this  the  hydrate  can  not  exist ;  but  it  can  be  in 
equilibrium  at  lower  temperatures  with  a  second  solution  containing 
a  larger  percentage  of  calcium  chloride  than  the  solid  crystals.  The 
vapor  pressures  of  these  solutions  have  been  determined  as  far  as  the 
point  D,  this  temperature  being  29.2°.  Strictly  speaking,  the  part 
of  the  curve  HCD  represents  a  labile  equilibrium,  the  system  being 
instable  with  respect  to  CaCl24H2O# ;  the  solution  curve  for  which, 
HK,  intersects  at  H,  29.8°,  but  this  modification  never  appears 
spontaneously  at  this  point,  and  there  is  therefore  no  difficulty  in 


Tav  Components 


-• 


following  the  curve  BOD  to  the  point  D  where  the  other  modification 
with  four  of  water,  CaClyjHX)/*,  appears.  The  saturated  solution 
of  CaClyiH^O/f  is  instable  with  respect  to  the  a  modification  along 
its  whole  length  from  F  (38.4°)  to  as  low  temperatures  as  it  has 
been  followed,  about  20°.  Below  29.2°  the  solution  is  also  instable 


•• 


FIG.  ii. 

with  respect  to  CaCl^^O.  The  solution  saturated  with  respect  to 
CaCly^HjOif  can  be  obtained  by  the  spontaneous  change  of  the  ft 
modification  and  is  stable  from  H  to  K,  from  29.8°  to  45.3°.  Be- 
low the  former  temperature  it  is  instable  with  respect  to  crystals  of 
the  hexahydrate ;  while  above  the  latter  the  crystals  lose  water 
changing  into  CaCl^HjO.  The  new  curve  KT,  has  a  maximum  vapor 
pressure  at  173°.  At  175.5°  this  s311  gives  way  to  CaCLHjO.  while 
the  undetermined  curve  for  the  anhydrous  salt  begins  at  about  260°. 
When  we  come  to  the  monovariant  systems  composed  of  two  solid 
phases  and  vapor  we  find  that  CaCl/iHjO  can  exist  in  equilibrium 
with  either  CaClJ4HJOor  or  CaClytHjO/S,  there  being  different  pres- 
sures in  the  two  cases.  This  is  interesting  because  it  shows  the 
very  definite  effect  exerted  by  the  other  solid  phase.1  It  seems  a 
truism  to  say  that  the  equilibrium  pressure  is  different  when  the  sub- 


Bancroft.  Phvs.  Rev.  3,  406  ( 1896). 


74  The  Phase  Rule 

stances  entering  into  equilibrium  change  ;  but  it  has  not  always  been 
clear  that  the  effloresced  salt  really  had  any  influence.  The  vapor 
pressures  for  the  system,  CaCl;6H2O,  CaCl24H2O.r  and  vapor,  are 
higher  than  the  values  at  the  same  temperatures  for  the  system, 
CaCl26H2O,  CaCl.,4H2Oj8  and  vapor,  and  the  latter  system  is  there- 
fore instable  with  respect  to  the  former.  The  curve  NK  represents 
the  stable  monovariant  system,  CaCl24H2O.v,  CaCl22H,O  and  vapor, 
while  the  corresponding  curve  for  the  system  containing  CaCl24H2O/3 
instead  of  the  a- modification  is  too  instable  to  be  determined.  All 
that  can  be  said  about  it  is  that  it  will  lie  below  the  curve  NK  except 
in  the  immediate  neighborhood  of  the  point  F.  The  curve  PL  for 
the  dihydrate,  monohydrate  and  vapor  presents  nothing  new.  The 
curves  for  the  monovariant  systems  composed  of  a  liquid  and  two 
solid  phases  were  not  studied  ;  but  they  start  from  the  quadruple 
points,  B,  H,  D,  F,  K,  L  and  M,  and  are  represented  approximately 
in  Fig.  9  by  the  lines  BB,,  HH,,  DD,,  KK,,  LLt  and  MM,.  We 
shall  get  a  clearer  idea  of  the  subject  if  we  consider  the  concentra- 
tion-temperature diagram,  Fig.  12.  The  distance  between  the  two 
chief  ordinates  represents  one  hundred  formula  weights  of  the  mix- 
ture, one  hundred  and  eleven  grams  of  calcium  chloride  being 
equivalent  to  eighteen  grams  of  wrater.  The  letters  have  tlie  same 
significance  as  in  the  preceding  three  figures.  Starting  at  o°  with 
pure  ice  and  anhydrous  calcium  chloride,  the  curve  AB  represents 
the  temperatures  and  concentrations  at  which  the  monovariant  sys- 
tem, ice,  solution  and  vapor  exists.  At  -55°  the  solution  becomes 
saturated  with  respect  to  CaCl26H3O,  and  we  have  the  non variant  sys- 
tem, ice,  CaCl.j6H.jO,  solution  and  vapor.  Further  addition  of  an- 
hydrous calcium  chloride  causes  the  ice  to  disappear  with  increased 
formation  of  the  hexahydrate,  and  the  curve  BH  represents  the  sta- 
ble equilibrium  between  CaCl26H3O,  solution  and  vapor.  At  29.8° 
the  solution  is  saturated  with  respect  to  CaCl.^HjOa',  and  if  a  crystal 
of  this  modification  be  added  we  shall  have  the  nonvariant  system, 
CaCl26HaO,  CaCJ^j-HjOa,  solution  and  vapor.  This  change  does  not 
take  place  spontaneously,  as  has  been  already  stated,  and  it  is  possi- 
ble to  follow  the  curve  BH  through  C  to  D.  At  this  latter  tempera- 
ture, 28.2°,  we  have  the  labile  but  easily  realized  nonvariant  system 
CaCl26H2O,  CaCl,4H2O£,  solution  and  vapor.  At  C,  30.2°,  we  have 
the  fusion  point  of  the  hexahydrate.  At  this  temperature  the  solution 


Two  Components 


75 


has  the  same  composition  as  the  solid  phase  with  which  it  is  in  equilib- 
rium. This  is,  therefore,  a  true  melting  point  for  the  hydrate,  while 
the  point  H  corresponds  to  the  inversion  point  for  NaJSO4ioHIO. 
From  C  to  D  we  have  the  first  example  of  a  solution  containing  more 
of  a  non- volatile  solute  than  the  compound  which  crystallizes  from  it. 


,-•        ;, 


\    / 


Fig.  12. 

This  phenomenon  had  already  been  observed  by  Roozeboom  for  com- 
pounds where  both  the  components  were  volatile.1  Although  the 
curve  CD,  like  HC,  is  instable  with  respect  to  CaCl^H.O.r,  it  is  en- 
tirely stable  with  respect  to  CaCl^HjO,  and  has  been  called  a  stable 
supersaturated  solution.  If  CaCl,6HjO  were  to  precipitate  from  it 
the  solution  would  become  more  concentrated  and  would  have  a  les- 
ser vapor  pressure.  Since  the  vapor  pressure  of  the  solution  is  the 
lowest  under  which  CaCl?6H,O  can  exist  at  that  temperature  without 
efflorescing,  it  is  clear  that  the  hexahydrate  can  not  precipitate  wnth- 
out  decomposition  or,  in  other  words,  can  not  precipitate.  The 
curve  DC  is  to  be  considered  as  the  prolongation  of  the  curve  JD  for 


1  Rec.  trav.  chem.,  4,  342  (1885). 


76  The  Phase  Rule 

the  monovariant  system,  CaCl26H2O,  CaCl24H,C0  and  vapor.  The 
point  has  been  raised  whether  the  curve  BHCD  is  continuous  or 
whether  it  is  composed  of  two  curves,  BHC  and  CD.  The  former 
view  is  supported  by  Roozeboom,1  the  latter  by  L,e  Chatelier.*  It  is 
still  open  to  any  one  to  take  either  side.  DF  is  the  curve  for  solu- 
tions saturated  with  respect  to  CaCl24H2O/2.  Below  D,  29. 2° ,  the  satu- 
rated solution  contains  more  calcium  chloride  than  the  solutions  sat- 
urated at  the  same  temperatures  with  respect  to  either  CaCl26H2O  or 
CaCl24H2O.'*,  and  is  therefore  instable  with  respect  to  both  these  salts. 
From  D  to  F  the  solution  is  instable  with  respect  to  the  less  soluble 
salt,  CaCl24H2O.i'.  At  F,  38.5°,  there  can  exist  the  labile  uonvariant 
system,  CaCl24H2O/2,  CaCl22H2O,  solution  and  vapor,  instable  with 
respect  to  CaCl24H2O'i/.  The  (3  modification  corresponds  thus  to 
Na2SO47H2O  in  that  it  is  never  in  a  state  of  stable  equilibrium  ;  but 
differs  qualitatively  because  the  intersections  of  its  solubility  curve 
with  the  next  higher  and  lower  hydrate  can  be  realized  experiment- 
ly.  The  solubility  curve  for  CaCl24H2O.'*  has  been  followed  as  far  as 
K,  45.3°,  where  it  meets  the  solubility  curve  for  the  dihydrate  ;  in 
the  other  direction  it  has  been  determined  as  far  as  20° ,  but  the  part 
of  the  curve  beyond  H  is  instable  with  respect  to  CaCl26H2O.  The 
solubility  curve  of  the  dihydrate  is  instable  with  respect  to  CaCl2 
4H2O*  from  F  to  K.  KL,  represents  the  stable  portion  terminated  at 
175.5°  by  the  appearance  of  the  monohydrate.  This  latter  salt  is 
replaced  by  the  anhydrous  salt  at  the  point  M,  somewhere  about 
260°,  and  beyond  this  the  experiments  do  not  go.  In  Tables  XII— 
XIV  are  the  data  for  concentrations  and  pressures  at  different  tem- 
peratures. The  wavy  line  in  the  expression  I^O^-rCad,,  denotes 
a  solution  containing  x  reacting  weights  of  calcium  chloride  dis- 
solved in  one  of  water.3  The  pressures  are  given  in  millimeters  of 
mercury  and  the  temperature  in  Centigrade  degrees  ;  x^  denotes  re- 
acting weights  of  calcium  chloride  per  reacting  weight  of  water  ;  x2 
reacting  weights  of  water  per  reacting  weight  of  calcium  chloride, 
and  x^,  reacting  weights  of  calcium  chloride  in  one  hundred  reacting 
weights  of  the  solution. 

1  Comptes  rendus,  108,  744  1013  (1889). 

2  Ibid.,  108,  565,  Soi,  1015  (1889). 

3  This  is  not  the  way  Roozeboom  writes  it,  but  I  have  changed  his  nomen- 
clature to  make  it  agree  with  his  usage  in  the  papers  on  the  gas  hydrates. 


Taw  Components 
TABLE  XH 


Xonvariant  Systems 

Temp.      Pressure 

Ice,                        CaC1^6H,O,         ^OsaaxoogCaCU,  Vapor   —  55° 
CaCl,6H,O,          CaCl»4HaO/S,       HXteaso.  iSsCaClj,  Vapor   +  29.2 
CUOjSHp,          CaCUiELpo.        H^Oassot  i64CaCl^  Vapor         29.8 
GaCClH,O/F,       CaCl^eH.O,         ELQ5S5aoL2O7CaCl,,  Vapor        38^4  1 
CaCMHpa,        CaCLzHJO,          H,Oas=so.2iiCaCi:,  Vapor         45.3 
CaCMHp,          CaC^tt.0,            E^Ossax^Cad,.  Vapor       175.5 
CaCl^Hp,            CaCl*                   H2O====OL556CaCl.,  Vapor       260. 

5-67 

6.80 

11.77 
842. 

TABLE  XILT 

Temp. 

Pressure);     -rj      ••     J%         Je»    >jj 

Tempt 

P^ssore 

** 

^ 

-*» 

Ice, 

solution  and  vapor 

CaCl^H/). 

solution 

and  vapor 

0° 

4.630.000    *        o.o 

40° 

8-5 

0.208 

4-81 

17.2 

-  5 

3.060.01758.7      1.6 

-:  : 

ii.  8 

O.2II 

4-73 

17.4 

—  IO 

2.030.02836.3      :  - 

50 

15-5 

0.215 

4-66 

17.7 

—  20 

0.04422.8      4.3 

55 

20.5 

0.218 

17-8 

—30 

0.05418.4     5.1 

60 

26.5 

O.222 

4-51 

18.1 

—40 

0.06316.0     5.8 

65 

34-0 

0.226 

4-43 

18.5 

—55 

0.069  14  5      6.4 

70 

43-0 

O.229 

4-37 

18.7 

Cad,6H,O,  solution  and  vapor 

75    i 

54-0 

0.232 

4^31 

18.9 

—55 

0.069  14.5     6.4 

So 

66.5 

0.236 

4-24 

19.1 

—25 

0.081  12.3     7.5 

85 

82.5 

O.240 

4.16 

19.4 

—  10 

0.970.08911.2      8.2 

100. 

0245 

4.08 

iy.6 

0 

1.940.09610.4     8.8 

100 

145- 

0.256 

3-90 

20.4 

IO 

3.460.105    9.59   9.6 

no 

204. 

0.269 

3.72 

21.  1 

20 

5.620.121    8.2810.9 

125 

326. 

0.286 

3-50 

22.2 

25 

6.700.133    7.5211.7 

135 

435- 

0-301 

3-33 

23.1 

28.5 

7.020,147    6.8  1  12.9 

140 

497- 

0-310 

3-23 

23-6 

295 

6.910.155    6.4613.5 

155 

680. 

0.348 

:  - 

259 

30.2 

6.700.167   6.0014.3 

160 

744- 

0.361 

2-77 

26.5 

29.6 

5-830.175.5.7014.9 

165 

790. 

0-383 

2.61 

27-7 

29-2 

5.670.185    5.41  15.6 

170 

834- 

0.413 

2.42 

29.2 

CaClr4H1O<T,solution  and  vapor 

175-5 

842. 

0.483 

2.07 

32-5 

20     f 

4.740.148   6.7812.9 

CaCl^O,  solution 

and  vapor 

25    i 

5.720.156   6.4213.5 

:75.= 

842. 

0.483 

2.07 

32-5 

29.8 

6.800.164  6.1014.1 

180 

910. 

0.488 

2.05 

32-7 

35 

8.640.174  5-7514-7 

185 

1006. 

0.490 

2.04 

32-9 

40  ; 

10370.187   5.34158 

190 

1114. 

0-495 

2.  02 

33-2 

45-3 

11.770.211    4-7317-4 

195 

1230. 

0.500 

2.OO 

33-3 

CaCUB 

,O/?,  solution  and  vapor 

200 

1354- 

0-505 

1.98 

33-5 

20 

3.560.170  5.9014.5 

205 

1491- 

0.510 

1.96 

33-8 

-  = 

4.640.177    5.6614.9 

235 

0-538 

1.86 

35-0 

29.2 

5-670.185    5.41  15.6 

260 

0-556 

1.8 

35-7 

30 

5.830.185    540157 

35 

7.130.199.  5.0416.6 

38.4 

7.800.207   4.8317.1 

7S 


The  Phase  Rule 
TABLE  XIV 


Temp. 

Pressure 

Temp. 

Pressure 

Temp. 

Pressure 

Temp. 

Pressure 

CaCl26H.2O,CaCl24H2Oa 

CaCU6H20,  CaCl«4H20/3 

CaCl24H2Oa,  CaCl22H2O 

CaCl22H20,  CaCloHoO 

Vapor 

Vapor 

Vapor 

Vapor 

-15° 

0.27         -15° 

0.22 

-15° 

0.17 

65° 

842 

0 

0.92              o 

0.76            o 

0-59 

78 

13 

10 

1.92 

10 

1.62 

10 

1-25 

IOO 

24 

20 

3-78 

20 

3-15 

20 

2.48 

129 

60 

25 

5.08 

25 

4-32 

25 

3-40 

155 

175 

29.8 

6.80 

29.2 

5-67 

30 

4.64 

165 

438 

35 

6.26 

170 

607 

40 

8-53 

175-5 

7J5 

45-3 

11.77 

842 

While  the  system,  calcium  chloride  and  water,  has  brought  out 
several  points  which  did  not  occur  in  the  system,  sodium  sulfateand 
water,  such  as  the  stable  existence  of  different  hydrates  under  suita- 
ble conditions,  the  existence  of  a  pressure  maximum  and  of  a  true 
fusion  point,1  yet  the  system  CaCl26H2O,  solution  and  vapor,  repre- 
sents a  labile  equilibrium  at  the  melting  point  of  the  hydrate,  and  it 
will  therefore  be  well  to  study  a  system  in  which  the  melting  salt  is 
in  stable  equilibrium  with  the  solution  and  vapor.  This  is  the  more 
necessary  since  the  difference  between  the  behavior  of  the  hydrates 
of  sodium  sulfate  and  the  hexahydrate  of  calcium  chloride  is  merely 
that  the  labile  equilibrium  can  be  realized  in  the  second  case  and  not 
in  the  first.  The  equilibrium  between  ammonium  bromide  and  am- 
monia2 with  the  three  solid  phases,  NH4Br,  NH4BrNH3  and  NH4Br 
3NH8  illustrates  only  the  points  already  brought  out ;  but  the  sys- 
tem, ferric  chloride  and  water,3  is  worth  considering  in  detail.  In 
the  concentration-temperature  diagram,  Fig.  13,  three  hundred  and 
twenty- five  grams  of  ferric  chloride  are  equivalent  to  eighteen  grams 
of  water. 


1  Other  salts  with  true  melting  points  are  ZnCl23H,.O,  NaH2PO32^H2O,  and 
the  hydrate  of  Al2Br6.  Cf.  Roozeboom,  Zeit.  phys.  Chem.  IO,  487  (1892). 

2Roozeboom,  Recueil  Trav.  Pays-Bas,  4,  361  (1885)  ;  Zeit.  phys.  Chem.  2, 
460  (1888). 

:!  Roozeboom,  Zeit.  phys.  Chem.  10,  477  (1892). 


Two  Components 


79 


The  solid  phases  which  occur  are  ice,  FesClsi2H2O, 

FejCVlH/)  and  anhydrous  ferric  chloride.  AB  is 
the  fusion  curve  with  ice  as  solid  phase.  At  B,  —55°,  the  so- 
lution becomes  saturated  with  respect  to  Fe,Cl,i2HaO,  forming  the 
nonvariant  system,  ice,  Fe^C^iaHjO,  solution  and  vapor.  BCDX  is 
the  solubility  curve  for  Fe.Cl.izH.O.  At  C,  37°,  the  solution  has 
the  same  composition  as  the  hydrate,  and  this  temperature  is  the 
melting  point  of  FeJCl4i2HIO.  From  C  to  X  the  saturated  solution 


FIG.   13. 

contains  more  ferric  chloride  than  the  crystals.  As  the  hydrate, 
FejCljBLjO,  can  appear  at  D,  27.4°,  the  part  of  the  curve  DX  repre- 
sents a  state  of  labile  equilibrium,  instable  with  respect  to  the  hydrate 
with  seven  of  water.  At  D  there  can  coexist,  FejCl^HjO,  Fe,Cl. 
/HjO.  solution  and  vapor.  The  curve  ODEFP  is  the  solubility  curve 
for  FejClj/HjO,  the  parts  OD  and  FP  representing  labile  equilibrium. 
The  hydrate  melts  at  32. 5° ,  shown  in  the  diagram  by  the  point  E.  At 
F,  30°,  there  is  the  monovariant  system,  Fe,Cl47H,O, 


8o  The  Phase  Rule 

solution  and  vapor.  MFGH  is  the  solubility  curve  for  the  hydrate 
Fe2Cl65H2O,  with  a  fusion  point  at  G,  56°.  The  labile  portion  of 
this  curve,  FM,  intersects  the  curve  DN  at  about  15°,  and  at  this 
temperature  there  can  exist  the  labile  non variant  system,  Fe.2Cl6 
i2H2O,  Fe2Cl65H2O,  solution  and  vapor,  instable  with  respect  to 
Fe2Cl67H2O.  At  55°,  H,  there  exists  the  stable  nonvariant  system 
with  Fe2Cl65H2O  and  Fe2Cl64H2O  as  solid  phases  and  at  K,  66°,  an- 
other with  Fe2Cl64H2O  and  anhydrous  ferric  chloride.  RHIK  rep- 
resents the  solubility  curve  for  Fe2Cl64H2O,  the  fusion  point  I  coming 
at  73-5°-  From  K  to  I/ the  solution  is  in  equilibrium  with  anhy- 
drous salt  ;  but  this  curve  has  not  been  followed  beyond  100°,  owing 
to  the  partial  decomposition  of  the  ferric  chloride.  In  the  case  of 
ferric  chloride  and  water,  there  are  four  hydrates,  each  of  which  can 
exist  in  stable  equilibrium  with  solutions  containing  more  salt  than 
the  crystals  and,  in  consequence,  each  of  the  four  is  in  stable  equi- 
librium at  its  melting  point.  It  is  worth  noticing  that  the  hydrate, 
Fe2Cl67H2O,  is  in  stable  equilibrium  only  over  a  very  narrow  range 
of  temperatures  and  concentrations,  and  possibly  never  would  have 
been  discovered  except  by  a  careful  study  of  the  solubility  curves. 
As  a  matter  of  fact,  Roozeboom  noticed  that  solutions  saturated  with 
respect  to  Fe2Cl65H2O  sometimes  solidified  entirely  at  the  temperature 
and  concentration  corresponding  to  F,  while  solutions  saturated  in 
respect  to  Fe2Cl6i2H2O  showed  the  same  phenomenon  under  the  cir- 
cumstances represented  by  D.  From  the  Phase  Rule  he  knew  it  was 
impossible  that  the  solid  phases  should  be  the  same  in  the  two  cases, 
and  he  was  thus  led  to  discover  the  hydrate  with  seven  of  water.  In 
this  particular  case  the  two  temperatures  are  nearly  three  degrees 
apart,  but  it  is  conceivable  that  this  difference  might  be  zero,  and 
then  only  the  solubility  determinations  would  show  the  existence  of 
a  hitherto  unknown  compound.  The  diagram,  Fig.  13,  enables  us 
to  predict  the  behavior  of  a  ferric  chloride  solution  if  evaporated  to 
dryness  at  a  constant  temperature  of  about  31°.  The  solution  would 
first  solidify  to  Fe2Cl6i2H2O,  become  liquid,  solidify  to  Fe2Cl67H2O, 
become  liquid  yet  again  and  solidify  for  the  third  time  with  forma- 
tion of  Fe2Cl65H2O.  Further  abstraction  of  water  would  cause  this 
last  hydrate  to  effloresce  in  the  usual  manner.  This  will  be  clear  if 
one  follows  the  horizontal  dotted  line  YY.  All  these  different  solids 


Two  Components  81 

and  solutions  represent  states  of  stable  equilibrium.  In  other  words, 
the  portion  of  the  line  YY  which  is  bounded  by  EF  and  FG  repre- 
sents a  solution  stable  with  respect  to  FejCljH.0  and  to  Fe^iaH^O, 
although  it  contains  more  of  the  solute  than  the  solutions  saturated 
with  respect  to  either  of  these  salts.  It  is  evidently  not  possible  to 
define  a  supersaturated  solution  as  one  which  contains  more  of  the 
solute  than  the  saturated  solution,  although  this  definition  would 
have  applied  to  all  the  solutions  studied  up  to  now,  with  the  excep- 
tion of  those  mixtures  of  calcium  chloride  and  water  in  the  field 
bounded  by  CHD  and  an  isothermal  line  through  C  in  Fig.  12. 
These  solutions  are  stable  with  respect  to  CaCl^H/X  As  almost 
the  whole  of  this  field  is  instable  with  respect  to  CaCly^Orr  the 
point  was  not  mentioned  in  the  discussion  of  the  equilibrium  between 
calcium  chloride  and  water,  it  being  reserved  until  a  more  striking 
illustration  of  the  phenomenon  could  be  found. 

All  solutions  between  the  temperatures  represented  by  C  and  D, 
Fig.  13,  which  contain  more  ferric  chloride  than  the  point  on  CD  for 
that  temperature  have  lower  pressures  than  the  minimum  pressure  at 
which  FejC^iaHjO  can  exist,  and  it  is  therefore  impossible  for  this 
salt  to  crystallize  from  the  solution.  The  same  reasoning  applies  to 
the  part  of  the  line  YY  bounded  by  EF  and  FG,  except  that  here  the 
vapor  pressures  are  lower  than  the  minimum  pressure  for  either 
FejCl^H/)  or  FejCljH/),  and  consequently  neither  can  crystallize 
from  the  solution.  These  are  cases  of  what  have  been  called,  for 
lack  of  a  better  name,  stable  supersaturated  solutions.1  It  does  not 
follow  that  a  hydrate  can  never  separate  from  a  solution  which  has  a 
lower  vapor  pressure  than  the  minimum  for  that  hydrate.  This  is 
true  only  in  case  the  solution  would  become  more  concentrated  by  the 
crystallization  of  the  hydrate  ;  in  other  words,  when  the  solution 
contains  more  of  the  solute  than  the  solid  phase  in  question.  If  the 
word  "  supersaturated  "  be  used  in  the  normal  sense  to  signify  a  so- 
lution unstable  in  presence  of  the  solid  phase,  with  respect  to  which 
it  is  said  to  be  supersaturated,  the  only  solutions  supersaturated  in 
respect  to  Fe,Cl,7H,O  are  those  in  the  field  OEP,  while  those  satu- 
rated in  respect  to  Fe^izHjO  are  bounded  by  the  curve  BCN.1  If 


1  Bancroft,  Jour.  Phys.  Chem.  I,  No.  3  (1896). 
•  Roozeboom,  Zeit.  phys.  Chem.  xo,  492  1  1892). 
6 


82  The  Phase  Rule 

CQ  be  a  line  perpendicular  to  the  abscissae,  all  solutions  to  the  left 
of  it  contain  less  ferric  chloride  and  all  to  the  right  of  it  more  ferric 
chloride  than  the  hydrate  Fe2Cl6i2H2O.  L,et  DS  be  an  isothermal 
line  passing  through  D  and  meeting  CQ  at  S.  It  is  then  possible  to 
predict  the  behavior  of  any  solution  supersaturated  with  respect  to 
Fe2Cl6i2H2O,  when  a  crystal  of  this  hydrate  is  added,  the  system  be- 
ing kept  always  at  constant  temperature.  From  any  solution  in  QCB 
the  system  passes  to  the  corresponding  solution  on  the  curve  BC  ; 
from  any  point  in  the  field  DSC,  it  passes  to  the  corresponding  point 
on  DC.  In  both  these  cases  the  monovariant  system  formed,  Fe2Cl6 
I2H.JO,  solution  and  vapor,  is  entirely  stable.  From  any  point  in 
the  field  NDSQ  there  will  be,  as  before,  precipitation  of  Fe2Cl6i2H2O 
and  formation  of  the  system  hydrate,  solution  and  vapor  ;  but  the 
solution  is  in  a  state  of  labile  equilibrium,  because  the  curve  DN  lies 
within  the  field  OEP  for  solutions  supersaturated  with  respect  to 
Fe2Cl67H?O.  The  final  stable  equilibrium  for  a  solution  having  the 
composition  represented  by  a  point  in  the  field  NDSA  will  be  the 
system,  Fe2Cl6i2H2O,  Fe2Cl67H2O,  and  vapor.  If  the  original  solu- 
tion were  represented  by  a  point  in  the  field  NDO,  a  crystal  of  either 
Fe2Cl6i2H2O  or  Fe2Cl6yH2O  would  cause  a  precipitation  of  the  salt 
added,  because  the  solution  is  supersaturated  with  respect  to  both 
hydrates ;  in  the  other  parts  of  the  field  NDSQ  the  hydrate  with 
twelve  of  water  must  crystallize  first,  and  addition  of  solid  Fe2Cl67H2O 
will  have  no  effect.  Similar  phenomena  take  place  in  the  supersat- 
urated regions  of  the  other  hydrates.  From  its  position  as  the  last 
hydrate  before  the  appearance  of  solid  solvent,  solutions  supersatu- 
rated with  respect  to  Fe2Cl6i2HgO  can  be  supersaturated  only  with 
respect  to  Fe2Cl67H2O,  while  the  supersaturated  region  OEP  for 
Fe2Cl67H2O  overlaps  that  for  Fe2Cl,i2H,O,  in  the  part  NDO,  and 
that  for  Fe2Cl65H2O,  in  the  part  PFM.  Bearing  this  in  mind,  there 
is  no  difficulty  in  applying  all  that  has  been  said  about  solutions  su- 
persaturated with  respect  to  Fe2Cl6i2H,O  to  solutions  supersaturated 
with  respect  to  the  other  hydrates.  The  best  definition  of  a  super- 
saturated solution  is  one  by  Budde,  quoted  by  Roozeboom  in  his  pa- 
per on  ferric  chloride.1  "  A  solution  is  supersaturated  with  respect 
to  a  definite  solid  phase  if,  in  contact  with  it  at  the  temperature  of 

1  Zeit.  phys.  Chem.  IO,  495  (1892). 


Two  Components  83 

the  experiment,  more  of  that  phase  separates.  Such  a  solution  con- 
tains more  salt  than  the  saturated  solution  if  the  latter  contains  more 
water  at  that  temperature  than  the  solid  phase,  and  vice-versa." 
This  is  put  in  the  following  form  by  Roozeboom  :  "A  solution  is  su- 
persaturated with  respect  to  a  solid  phase  at  a  given  temperature  if 
its  composition  is  between  that  of  the  solid  phase  and  the  saturated 
solution." 

Since  the  solution  at  D  is  more  concentrated  than  FejCl6i2H,O 
and  less  so  than  Fe2Cls7H2O,  it  follows  that  this  solution  will  solidify 
to  the  two  hydrates  without  change  of  temperature.  This  occurs 
only  in  case  the  composition  of  the  solution  lies  between  those  of  the 
two  solid  phases  and  is  not  a  general  phenomenon.  The  solution  in 
equilibrium  with  Na,SO4ioH2O  and  Na2SO4  contains  more  water 
than  either  salt,  and  the  same  is  true  of  the  solution  in  equilibrium 
with  CaCl^HjOor  and  CaCl,2H,O.  In  both  these  cases  the  solution 
will  deposit,  on  cooling,  the  salt  stable  at  lower  temperatures  and  the 
temperature  will  fall. 

In  Tables  XV-XVI  are  the  experimental  data  for  ferric  chloride 
and  water.  Under  the  heading  x^  are  the  concentrations  in  reacting 
weights  of  ferric  chloride  per  reacting  weight  of  water  ;  under  ,r,  the 
reciprocal  values,  and  under  .r3  the  reacting  weight  of  ferric  chloride 
in  one  hundred  reacting  weights  of  the  solution.  Under  the  head- 
ing "  non variant  systems,"  only  the  solid  phases  are  given,  solution 
and  vapor  being  understood  in  each  case. 

TABUS  XV 


Nonvariant  Systems 

Temp. 

Ice, 
Fe,Cl«i2HA 

Fe,Cl6i2H,0                -55° 
Fe.CljHp                      27.4 

Fe,CljHA 

Fe,Cl65H,0 

30- 

Fe,Clsi2HA 

Fe,C1.5H,0 

15- 

FesCl65HA 

Fe,Clg4H3O 

55- 

Fe,Cl«4HA 

Fe,Cl6 

66. 

Melting 

Points 

Fe,Cl6i2H,0 

37- 

Fe,Cl67H,0 

32-5 

Fe,Cls5H,0 

56. 

FejCl^HjO 

73-5 

The  Phase  Rule 
XVI. 


Temp. 

*i 

x* 

x* 

Temp. 

xv 

*> 

*, 

Ice,  solution  and  vapor 

Fe^Cl^HjO,  solution  and  vapor 

0° 

o.ooo 

00 

o.oo 

12° 

0.129 

7-77 

11.17 

—  10 

O.OIO 

100.              0.99             20 

0.139 

7.17 

12.25 

-20.5 

0.016 

61.      \     1.62 

27 

0.149 

6-73 

12.94 

-27-5 

0.019 

52.6 

1.87 

30 

0.151 

6.61 

13.12 

-40 

0.024 

42.2 

2.32 

35 

0.156 

6.40 

13-52 

—55- 

0.027 

36.4 

2.65 

50 

0.175 

5.7i 

14.90 

Fe.,Cl6i2H2O,  solution  and  vapor 

55 

0.192 

5.22 

16.07 

-55            0.027 

36.4 

2.65 

56 

0.20O 

5-oo 

16.67 

—41            0.028 

35-6 

2.74 

55 

0.203 

4.92 

16.91 

—27            0.030 

33-6 

2.89 

Fe2Cl64H2O,  solution  and  vapor 

0 

0.041 

24.2 

3.97 

50 

0.200 

5-Oi 

16.54 

10 

0.045 

22.  0 

4-35 

55 

0.203 

4.92 

16.91 

20 

0.051 

19.6 

4-85 

60 

0.207 

4.88 

17.14 

30 

0.059 

16.9 

5.60 

60 

0.215 

4.64 

17.70 

35 

0.068 

I4.8 

6-35 

72.5 

0.234 

4.28 

18.50 

36.5 

0.079 

12.6        7.34 

73-5 

0.250 

4.00 

20.00 

37 

0.083 

12.0 

7.69 

72.5 

O.262 

3-82 

2O.70 

36 

0.093 

10.8        8.49 

70 

0.279 

3-58 

21.83 

33 

0.105 

9-57 

9-25 

66 

0.292 

3-43 

22.60 

30 

O.II2 

8.92 

9.84 

Fe2Clt,  solution  and  vapor 

27.4 

0.122 

8.23    10.84 

66 

0.292 

3-43 

22.60 

20 

0.128 

7.8o|  11.38 

70 

0.294 

3-40 

22.7 

10 

0.132 

7.57 

11.67 

75 

0.289 

3.46 

22.45 

8 

0.137 

7-30 

12.05 

80 

O.292 

3-43 

22.6 

FezCl67H2O,  solution  and  vapor 

IOO 

0.298 

3-33 

22.9 

20 

O.II4 

8.81 

10.19 

27.4 

O.I22 

8-23 

10.84 

32 

0.136 

7-38 

11.94 

32-5 

0.143 

7.00 

12.50 

30 

O.I5I 

6.61 

13.12 

| 

25 

0.155 

6.47 

13.45 

j 

While  it  has  been  possible  to  realize  some  states  of  labile  equi- 
librium in  the  systems  which  have  been  under  discussion,  it  has  not 
been  easy  to  follow  the  curves  more  than  a  short  distance  beyond  the 
point  where  the  new  phase  could  appear.  Systems  differ  very  much 
in  respect  to  this.  It  has  already  been  mentioned  that  with  the  diva- 
riant  systems,  solution  and  vapor,  it  is  possible  to  obtain  marked 
supersaturation  with  some  solutes,  almost  impossible  with  others.  In 
the  same  way,  some  monovariant  systems  change  spontaneously  as 


T-apo  C&mpffHftib  85 

soon  as  the  equilibrium  for  the  nonvariant  system  is  reached,  while 
others  can  be  carried  far  beyond  it,  A  striking  example  of  the  latter 
is  to  be  found  in  the  hydrates  of  thorium  sulfate,  the  behavior  of 
which  has  been  studied  by  Bakhuis  Roozeboom.1  Fig.  14  is  the  coo 


•    * 


•••'    '"  •--*• 


_ 


= 


FIG.  14. 

centration-temperature  diagram  for  thorium  sulfate  and  water.  ABC, 
FG.  HK  and  DBE*  represent  the  solutions  saturated  with  respect  to 
thorium  sulfate  with  nine*  seven,  six  and  four  of  water,  respectively. 
It  is  dear  that  AB  and  BD  are  the  only  stable  solutions ;  but  the 
labile  forms  can  be  obtained  over  a  wide  range  of  temperatures. 
Even  in  presence  of  the  stable  modification  it  is  often  hours  and  even 
days  before  the  change  is  completed.  If  the  anhydrous  salt  be 
brought  in  contact  with  water,  it  will  often  dissolve  completely,  only 
precipitating  the  stable  hydrate  after  long  standing,  whereas  it  is 
usual  for  a  dehydrated  salt  to  take  up  water  at  once.3 


•  Zest.  phys.  Chen.  5.  198  ( 1890). 

«Th(SO4\4H2O  dissolves  with  evolution  of  heat.     The  letter  B  is  missing 
from  the  interaction  of  A  C  and  D  E. 

i  to  present  another  case  of  extra  - 
with  respect  to  them  are  so  improb- 
Unebargcr,  Am.  Cbem.  Jowr.  15.  »5 
--- 


CHAPTER  VI 

VOLATILE  SOLUTES 

It  has  been  assumed  hitherto  that  one  of  the  components  had  an 
immeasurably  low  vapor  pressure.  Under  these  circumstances  the 
vapor  phase  contains  only  one  component,  and  condensation  has  the 
effect  of  adding  more  of  that  component,  of  diluting  the  solution  if 
there  happens  to  be  a  liquid  phase.  If  both  components  are  volatile 
this  is  no  longer  true,  and  under  certain  circumstances  condensation 
means  adding  more  of  one  component,  and  under  others  the  reverse. 
With  one  non-volatile  component,  increase  of  external  pressure  causes 
the  monovariant  system,  solid,  solution  and  vapor,  to  pass  into  one  of 
the  two  divariant  systems,  solid  and  solution,  or  .solution  and  vapor, 
except  when  the  solution  contains  more  of  the  solute  than  the  solid 
phase.  When  both  components  are  volatile  this  is  no  longer  so,  and 
it  is  possible  to  pass  by  increase^!  pressure  from  solid,  solution  and 
vapor  to  solid  and  vapor.  This  may  be  illustrated  by  the  equilibri- 
um between  iodine  and  chlorine.1  The  solid  phases  which  occur  are 
iodine,  two  modifications  of  iodine  monochloride  and  iodine  trichlo- 
ride. Under  suitable  circumstances,  it  would  be  possible  to  have 
solid  chlorine  ;  but  this  does  not  come  within  the  range  of  the  pres- 
ent discussion.  The  vapor  pressures  of  the  labile  monochloride  IC1/3 
differed  so  little  from  those  of  Ida  that  they  could  not  be  deter- 
mined and  are,  therefore,  not  given  in  the  pressure-temperature  dia- 
gram, Fig.  15.  This  equality  applies  only  to  the  total  pressure.  It 
is  not  probable  that  the  partial  pressures  of  iodine  and  chlorine  vapor 
in  equilibrium  with  the  two  modifications  are  the  same,  but  only  that 
their  sum  is.  This  point  has  not  been  determined  experimentally. 

In  the  diagram  AB  .represents  I2,  Ida  and  vapor  ;  BC,  I2,  solu- 
tion and  vapor  ;  BDE,  Ida,  solution  and  vapor  ;  HE,  ICl^,  IC18  and 
vapor  ;  EF,  IC13,  solution  and  vapor.2  The  monovariant  systems 


1  Stortenbeker,  Zeit.  phy's.  Chem.  3,  n  (1889)  ;  XO,  183  (1892). 

2  In  order  to  keep  the  figure  a  convenient  size,  two  pressure  scales  are  used. 


Tteo  Components 


containing  two  solid  phases  and  solution  have  not  been  studied  ;  but 
they  ma\*  be  represented  approximately  by  the  dotted  lines  BB,  and 
EE,.  The  nonvariant  systems  are  L.  Ida,  solution  and  vapor  exist- 
ing at  B,  7.9°,  and  IC1«,  IC1,,  solution  and  vapor  possible  at  E,  22.7°. 
The  curve  CtC  represents  the  vapor  pressure  of  solid  iodine,  termi- 
nating at  114.2°,  the  melting  point  of  pure  iodine,  CC,  being  the 
curve  for  liquid  iodine  and  vapor.  Below  this  curve  solid  iodine  can 
not  exist.  It  will  be  noticed  that  the  vapor  pressure  of  the  system 


-'•?.'* 


i 


turn   tficklonde 


.--*;r< 


* 


FIG.  15. 

iodine,  solution  and  vapor,  is  higher  than  that  of  the  solid  solvent,  a 
phenomenon  which  can  only  occur  with  a  volatile  solute.  In  the 
neighborhood  of  100°  this  curve  has  a  maximum  vapor  pressure  an- 
alogous to  the  maximum  vapor  pressure  of  the  system,  Cad,6H,O, 
solution  and  vapor,  just  below  the  fusion  temperature  of  the  hexa- 
hydrate.  At  63.7°  the  vapor  pressure  of  the  system.  IC1,.  solution 
and  vapor,  becomes  equal  to  atmospheric  pressure  so  that  the  melt- 
ing point  of  the  compound  can  not  be  determined  in  open  vessels. 
There  exist  in  the  fields  QCBA,  ABDEH,  HEF  and  FEDBC,  the 
divariant  systems  composed  of  vapor  in  equilibrium  with  iodine, 


The  Phase  Rule 


nionochloride,  trichloride  and  solution,  respectively.  As  these  are 
the  more  interesting  systems  I  have  marked  them  on  the  diagram. 
The  other  divariant  systems,  iodine  and  nionochloride,  iodine  and 
solution,  monochloride  and  trichloride,  monochloride  and  solution, 
trichloride  and  solution,  exist  in  the  fields  B,BA,  BjBC,  E^H, 
BjBDEE!,  E,EF. 

If  we  start  with  a  very  large  quantity  of  iodine  and  chlorine  vapors 
at  the  pressure  and  temperature  represented  by  the  point  M,  and 
compress  the  mixture  at  constant  temperature,  the  changes  will  be 
represented  by  the  dotted  line  MMr  The  following  monovariant 
and  divariant  systems  will  be  formed  :  iodine  and  vapor  ;  iodine,  so- 
lution and  vapor  ;  solution  and  vapor  ;  nionochloride,  solution  and 


\ 


V/r 


Iodine 


FIG.  1 6. 

vapor  ;  monochloride  and  vapor  ;  monochloride,  solution  and  vapor  ; 
solution  and  vapor  ;  trichloride,  solution  and  vapor  ;  trichloride  and 
vapor  ;  trichloride,  solution  and  vapor  ;  solution  and  vapor.  This 
multitude  of  changes  is  due  largely  to  the  fact  that  both  the 
monochloride  and  the  trichloride  can  exist  in  stable  equilibrium 


Two  Components  89 

with  solutions  containing  more  and  less  chlorine  than  the  crystals 
themselves.  This,  as  well  as  the  place  occupied  by  Id/?  in  the 
equilibrium,  will  be  clearer  if  we  consider  the  compositions  of  the 
different  solutions  as  shown  in  the  concentration -temperature  dia- 
gram. Fig.  1 6.  One  hundred  and  twenty-seven  grams  of  iodine 
are  equivalent  to  thirty-five  and  one-half  grams  of  chlorine.  The 
composition  of  the  vapors  is  represented  as  far  as  possible.  The 
letters  have  the  same  significance  as  in  the  preceding  diagram.  The 
curves  for  the  vapors  have  the  same  letters  as  the  corresponding  so- 
lutions, but  underlined  to  distinguish  them.  The  melting  point 
of  Ida  is  found  to  be  at  27.2°,  KGL  is  the  solubility  curve  for  IC10, 
this  compound  melting  at  13.9^-  The  curve  represents  a  state  of 
labile  equilibrium  at  all  temperatures.  There  is  a  labile  nonvariant 
system  possible  at  K,  0.9°,  and  another  at  L.  12.0°.  At  the  first 
there  coexist  I,.  IC10,  solution  and  vapor :  at  the  second,  IC1£.  ICL,, 
solution  and  vapor.  The  continuation  of  the  curve  EF  represents 
the  solutions  in  equilibrium  with  IC1,.  and  containing  more  chlorine 
than  the  crystals.  This  curve  if  prolonged  terminates  at  about  -102°  „ 
when  solid  chlorine  begins  to  separate  from  the  solution.  There 
seems  to  be  no  question  from  the  experimental  data  that  the  two 
branches  of  the  solubility  curve  for  IClor,  BD  and  ED  meet  at  an  an- 
gle, but  this  is  denied  by  Stortenbeker  on  the  strength  of  a  thenno- 
dynamic  formula  by  van  der  Waals.  In  view  of  the  tacit  assump- 
tions which  have  been  so  often  discovered  in  thermodynamical  for- 
mulas, this  conclusion  can  hardly  be  considered  a  wise  one,  especial- 
ly as  we  are  not  yet  in  a  position  to  say  at  what  point  iodine  ceases 
to  be  solvent  and  becomes  solute,  nor  what  effect  this  will  have  on 
the  equations  of  equilibrium.  In  this  particular  case  it  is  assumed 
explicitly'*  that  the  solution  consists  of  a  mixture  of  iodine  and  chlo- 
rine, and  that  there  is  no  fused  monochloride  present.  This  may  or 
may  not  be  true.  There  is  no  way  at  present  of  determining  it.  In 
Tables  XYII-XYni  are  the  experimental  data  for  this  system.  The 
pressures  are  given  in  millimeters  of  mercury,  .r,  denotes  units  of 
chlorine  per  unit  of  iodine ;  xv  units  of  iodine  per  unit  of  chlorine  ; 
.r,,  units  of  chlorine  in  one  hundred  units  of  solution ;  jr,  units  of 
chlorine  per  unit  of  iodine  in  the  vapor.  The  values  of  r  are  only 
approximative. 


;  Zeit  phys.  Chem.  IO,  i^j  *•  1892). 


The  Phase  Rule. 
TABLE  XVII 


Non  variant  Systems 

Vapor 

Temp. 

Pressure 

Solid                   Liquid 

I2,  IC1" 
I2Cl/r,  Id, 

I«  0.66C1 
I—i.igCl 

I 

==0.92  Cl 
ssi.75Cl 

7-9°              ii 
22.7               42 

Melting  Points 


I,,  114.2°  ICla;27. 2°          IC1/8,  13.9°    I    Id,,  ioi' 

TABLE  XVIII 


Temp. 

Pressure 

* 

*. 

*\* 

Temp. 

Pressure 

I.*' 

y 

Iodine,  solution  and  vapor. 

ICltf,  IC1,,  vapor. 

5-0°                 0.69    1.4541 

9-7° 

16. 

7.9        ii.        0.66    1.51  40 

15- 

24- 

10.                      0.65 

1-5439 

19- 

32. 

20.               15.          0.54 

1-8535 

20. 

36. 

25.               20. 

22.7 

42- 

30.               25.          0.49 

2-0433 

IClj,   solution  and  vapor. 

40. 

43-       0-45 

2.22  31 

20. 

1.170.8554. 

50- 

63.       0.40 

2.5029 

22.7 

42. 

1.190.8454.   1  1.  7. 

70. 

0.28 

3-5722 

25- 

49-5 

[ 

100. 

114.2 

0.  10 

91.       p.oo 

10. 

00 

9 

30- 
40. 

72. 
147. 

1.260.79 

1-370.73 

56.     2.3< 
58.     4.2( 

ICltf,  solution  and  vapor 

50- 

296. 

7-9 

it.       0.66 

1.51 

400.92 

60. 

571- 

i-55 

0.65 

61.    7.0 

10. 

12.       0.69 

i-45 

4i 

64.  i 

773- 

13-5 

13-5 

70.1 

1183. 

15- 

TO. 

0.76    1.31 

430.93 

73-6 

1.66 

0.6063.     8.7 

20. 

22. 

0.84    1.19 

46 

78-7 

2284. 

22.5 

24-5 

85-3 

3549- 

27.2 

39- 

I.OO     I.OO 

50 

1.04 

89. 

2.020.5067. 

26.0 

41. 

90.4 

5190. 

25- 

41.5 

I.  II 

0.90 

53 

1.45 

95-4 

8i37- 

22.7 

42. 

1.19   0.84 

54 

i-75 

96. 

2.430.41 

71. 

JC1/8,  solution  and  vapor 

100.5 

11707. 

°-9 

10.72    1.4242 

101. 

16  Atm. 

3.00 

0-3375- 

7.0 

0.84    1.1946 

94- 

92. 

13-9 

i.oo    1.0050 

75- 

97-7 

12.0 

n.  10  0.91  52 

60.5 

99- 

Iodine,   IClfl'and  vapor. 

42-5 

99-3 

5-0          9- 

30. 

99-8 

7-9        ii- 

THY?  Components  91 

It  often  happens  that  a  volatile  solute  forms  no  solid  compounds 
with  the  solvent  and,  in  that  case,  the  only  monovariant  system  pos- 
sible will  be  solid  solvent,  solution  and  vapor,  provided  the  pressure 
be  not  increased  to  the  point  where  the  solute  condenses  to  a  liquid, 
and  provided  the  solvent  occurs  in  only  one  solid  modification.  The 
diagram  for  such  a  system,  the  solute  being  a  gas,  is  confined  to  the 
fusion  curve  CB,  Fig.  16.  For  each  pressure  there  is  a  definite  con- 
centration in  the  solution  phase  and  a  definite  temperature  at  which 
the  monovariant  system  can  exist.  The  pressure,  being  the  sum  of 
the  partial  pressures  of  the  solvent  and  solute,  is  usually  higher  than 
that  of  the  solid  solvent  at  the  same  temperature  ;  but  with  a  very 
soluble  gas  this  is  not  necessarily  true,  and  the  system  may  behave 
like  one  with  a  non-volatile  solute,  or  the  vapor  pressure  of  the  sys- 
tem may  even  pass  through  a  minimum. 

An  example  of  this  is  possibly  to  be  found  in  the  equilibrium 
between  hydrochloric  or  hydrobromic  acid  and  water,  though  this 
particular  curve  has  not  been  studied  by  Roozeboom.1  Neither  of 
these  instances  is  really  satisf acton,-,  since  both  hydrochloric  and  hy- 
drobromic acid  form  solid  compounds  with  water.  With  a  sparingly 
soluble  gas  the  increase  of  pressure  with  increasing  concentration 
may  be  very  great,  and  it  becomes  an  interesting  question  whether 
this  may  not  have  some  effect  on  the  equilibrium  between  solid  and 
solution,  considered  as  an  increase  of  pressure.  The  subject  has  not 
been  studied  carefullv-  from  this  point  of  view,  but  the  bulk  of  the 
evidence  points  to  the  conclusion  that  the  equilibrium  between  solid 
and  solution  or  solid  and  solid1  is  a  function  of  the  total  pressure  on 
the  solid  phase,  independent  of  the  fact  that  this  pressure  is  due  in 
part  to  one  of  the  components  in  the  vapor  phase.*  There  will  be  two 
influences  at  work  to  change  the  freezing  point  in  the  case  of  a  sys- 
tem composed  of  solid  solvent,  solution  and  vapor.  There  will  be 
the  change  in  the  partial  pressure  of  the  solvent  due  to  the  concen- 
tration of  the  solute  in  the  solution  phase.  This  always  lowers  the 
freezing  point  when  the  solid  phase  is  pure  solvent.  In  addition 


1  Recueil  Trav.  Pays-Bas,  3,  84  (1884) ;  ZeiL  phys.  Chem.  2,  454  (1888). 
*Cf-  Reicher,  Recueil  Trav.  Pays-Bas.  a,  246  ( 1883). 

1  For  numerical  data  in  regard  to  the  freezing  point  of  water  when  saturated 
ith  different  gases  at  atmospheric  pressure,  see  Prytz.  Beiblatter  19,  870  (1895). 


92  The  Phase  Rule 

there  is  the  increased  pressure  due  to  the  concentration  of  the  solute 
in  the  vapor  phase.  This  will  lower  the  freezing  point  if  the  solvent 
is  less  dense  as  solid  than  as  liquid,  which  is  the  case  with  water  ;  it 
will  raise  the  freezing  point  in  case  the  solvent  contracts  in  freezing, 
as  most  solvents  do.  In  the  first  case  the  two  forces  work  together 
giving  a  greater  depression  of  the  freezing  point  than  would  be  cal- 
culated from  the  concentration  alone  ;  in  the  second  case  they  act  in 
opposite  directions.  It  is  quite  possible  that  with  a  substance  which 
expands  in  melting  and  a  gas  almost  insoluble  in  the  liquid  solvent, 
the  effect  due  to  pressure  might  eventual^  overbalance  the  effect  due 
to  solubility  ;  in  which  case  the  freezing  point  would  sink,  pass 
through  a  minimum  and  then  increase.  This  has  never  been  ob- 
served experimentally.  The  hypothetical  case  that  the  volume 
change  might  have  different  signs  when  the  solvent  separates  as  solid 
from  the  pure  liquid  or  from  the  solution,  is  too  complicated  to  be 
worth  considering. 

Of  more  importance  is  the  behavior  of  the  di variant  system, 
solid  and  vapor.  Here  there  can  be  any  pressure  at  any  tempera- 
ture within  the  limits  for  the  appearance  of  new  phases  ;  but  the 
question  suggests  itself  whether  the  second  or  gaseous  component 
has  any  effect  upon  the  partial  pressure  of  the  solid.  It  is  held  that 
there  is  no  such  influence  ; '  but  this  does  not  seem  entirely  satisfac- 
tory.2 There  are  cases  of  increased  volatility  on  the  part  of  the  solid 
which  do  not  seem  to  be  explicable  by  assuming  the  formation  of 
volatile  compounds.  Instances  of  this  are  the  behavior  of  iodine  in 
the  presence  of  carbonic  acid  ; 3  of  potassium  iodide  in  alcohol  vapor  ;  * 
of  boric  acid  and  methyl  alcohol ; 5  of  zinc  oxide  and  sulfite  in  pres- 
ence of  zinc  vapor.6  Instances  where  there  may  easily  be  formation 
of  volatile  compounds  but  where  it  is  doubtful  whether  that  explana- 

1  Qstwald,  Anal.  Chem.  33  ;  Nernst,  Theor.  Chem.  306. 

2  Bancroft,  Jour.  Phys.  Chem.  I,  No.  3  (1896). 
sVillard,  Comptes  rendus,  I2O,  182  (1895). 

4  Hannay,  Proc.  Roy.  Soc.  30,  178  (1880). 

5  Gooch,  Proc.  Am.  Acad.  22,  167  (1886).     Cf.  also  Margueritte-Delachar- 
lonnay,  Comptes  rendus,  103,  1128  (1886)  ;  Bailey,  Jour.  Chem.  Soc.  65,  445 
(1894). 

6  Morse  and  White,  Am.  Chem.  Jour.  XI,  258,  348  (1889). 


Two  Components  93 

tion  is  sufficient  are  the  volatility  of  nickel  chloride  in  a  current  of 
hydrogen  and  of  iron,  nickel  and  zinc  in  a  current  of  hydrochloric 
acid.1  In  these  last  cases  we  have  more  than  two  components.  The 
volatility  of  platinum  in  presence  of  chlorine,  hydrogen  or  nitrogen* 
under  proper  circumstances  is  a  very  interesting  phenomenon  though 
very  possibly  having  no  bearing  on  the  question  under  consideration. 
The  point  at  issue  is  whether  a  vapor  phase  containing  two  or  more 
components  is  a  solution  or  a  mixture.  If  the  change  from  a  liquid 
to  a  vapor  is  a  continuous  one,  the  vapor  must  possess  a  solvent 
power  though  it  may  be  to  an  infinitesimal  degree.  If  we  accept 
this  we  are  forced  to  conclude  that  a  vapor  or  gas  always  has  an 
effect  upon  the  partial  pressure  of  a  solid  with  which  it  forms  no 
compounds.  The  partial  pressure  of  the  solid  will  increase  if  the 
solid  is  soluble  in  the  other  vapor  or  gas  and  will  decrease  if  the  re- 
verse is  true.  While  this  effect  may  fall  within  the  limits  of  experi- 
mental error  in  many  cases,  it  is  extremely  improbable  that  it  does 
so  always. 

A  different  nonvariant  system  from  any  yet  considered  is  one  in 
which  there  is  equilibrium  between  two  solid  modifications  of  the 
solute,  solution  and  vapor.  Sulfur  crystallizes  in  the  monoclinic 
form  from  a  hot  solution  in  toluene,  in  the  rhombic  form  at  lower 
temperatures.  From  this  it  follows  that  at  some  intermediate  tem- 
perature the  two  modifications  can  exist  simultaneously  in  stable 
equilibrium  with  the  solution  and  vapor.  So  far  as  we  know  this 
temperature  is  identical  with  the  inversion  point  when  no  solvent  is 
present.1 


1  Schiitzenberger,  Comptes  rendns,  113,  177  ( 1891 ). 
-  Lehmann,  Moleknlarphysik,  II,  72. 
J  Cf.  Nernst,  Theor.  Chem.  505. 


CHAPTER  VII 


TWO   IvIQUID   PHASES 

There  are  many  substances  which  are  not  miscible  in  all  propor- 
tions in  the  liquid  state,  and,  with  two  components  of  this  class,  it  is 
possible  to  have  two  liquid  phases,  one  containing  an  excess  of  one 
of  the  substances,  the  other  of  the  other.  The  component  which  is 
in  excess  is  in  all  cases  the  solvent,  the  other  the  solute  and  it  is  cus- 
tomary to  speak  of  one  of  the  liquid  phases  as  a  solution  of  A  in  B 
and  of  the  other  as  a  solution  of  B  in  A.  As  an  example  of  this 
class  we  will  take  naphthalene  and  water.  Fig.  17  gives  an  approxi- 
mate representation  of  the  change  of  concentration  with  the  tempera- 
ture. The  diagram  lays  no  claim  to  accuracy  because  the  solubili- 
ties have  not  been  determined.  AB  is  the  fusion  curve  for  naphtha- 


FIG.    17. 

lene  in  the  presence  of  water.  Naphthalene  is  the  solvent  and  water 
is  the  solute,  the  latter  lowering  the  freezing  point  of  the  former. 
When  the  concentration  represented  by  B  is  reached,  further  addi- 
tion of  water  brings  about  no  further  depression  of  the  freezing 
point.  Instead,  there  appears  a  second  liquid  layer  having  the  con- 
centration represented  by  the  point  D,  forming  the  non variant  sys- 


Two  Components  95 

tern,  solid  naphthalene,  solution  of  water  in  naphthalene,  solution 
of  naphthalene  in  water,  and  vapor.  The  temperature  at  which 
this  is  possible  is  about  74°.  This  system  can  exist  only  at 
one  temperature  and  one  pressure.  Any  change  in  the  conditions  of 
equilibrium  brings  about  the  disappearance  of  one  of  the  phases  be- 
fore either  temperature  or  pressure  can  change.  If  the  solid  phase 
be  made  to  disappear  we  have  the  monovariant  system,  two  liquid 
phases  and  vapor.  For  a  given  temperature  the  pressure  of  the  sys- 
tem and  the  concentrations  in  the  two  liquid  phases  are  entirely  de- 
termined and  addition  of  either  water  or  naphthalene  produces  a 
change  in  the  relative  masses  of  the  phases,  not  in  their  composi- 
tions. These  latter  are  represented  by  the  curves  BC  and  DE.  It 
would  be  possible  to  take  either  curve  as  representing  the  mono- 
variant  system  and  disregard  the  other  but  it  conveys  more  informa- 
tion to  have  both  in  the  diagram,  and  the  points  at  which  an  isother- 
mal line  cuts  the  two  curves  show  the  concentrations  of  the  two 
solutions  in  equilibrium. 

Since  the  two  solutions  and  the  vapor  constitute  a  monovariant 
system  they  must  have  a  constant  boiling  point  so  long  as  the  exter- 
nal pressure  remains  constant.  This  is  found  to  be  true  experiment- 
ally, the  boiling  mixture  distilling  at  constant  temperature — in  this 
case  at  about  95° — so  long  as  the  two  liquid  phases  are  present. 

It  is  to  be  noticed  that  there  are  two  sets  of  solutions  repre- 
sented by  BA  and  BC  in  both  of  which  naphthalene  is  solvent.  The 
first  is  a  fusion  curve,  the  second  a  solubility  curve.  The  distinc- 
tion between  them  is  that  in  the  latter  case  the  solution  is  saturated 
in  respect  to  the  solute,  in  the  former  in  respect  to  the  solvent. 

If  a  mixture  of  naphthalene  and  water  forming  two  liquid 
phases  and  vapor  be  allowed  to  cool  at  constant  volume,  the  temper- 
ature will  fall  until  the  solid  naphthalene  begins  to  appear  at  74°. 
The  temperature  will  remain  constant  until  the  whole  of  the  liquid 
phase,  solution  of  water  in  naphthalene,  has  disappeared  leaving  the 
monovariant  system,  solid  naphthalene,  solution  of  water  in  naph- 
thalene, and  vapor.  The  temperature  falls  again,  the  system  passing 
along  the  line  DF  until  at  F  the  cryohydric  point  is  reached  and  ice 
separates,  forming  the  nonvariant  system,  naphthalene,  ice,  solution 
of  naphthalene  in  water  and  vapor.  The  temperature  now  remains 


,,6 


The  Phase  Rule 


constant  until  the  whole  of  the  solution  has  disappeared.  DF  is  a 
continuation  of  ED,  the  break  in  the  curve  at  D  being  caused  by  the 
change  in  the  heat  of  solution  when  the  solute  separates  in  the  solid 
state.1  In  the  solutions  represented  by  ED  and  DF,  water  is  solvent 
and  naphthalene  is  solute.  The  curve  FH  is  the  fusion  curve  for 
ice.  The  complete  concentration-temperature  diagram  for  two  sub- 
stances, which  do  not  form  any  compounds  and  which  do  form  two 
liquid  phases  at  some  temperature,  consists  of  two  fusion  curves  and 
two  solubility  curves.  In  the  fields  ABC  and  EFH  there  exist  the 
divariant  systems,  solution  and  vapor.  In  the  first,  naphthalene  is 
solvent ;  in  the  second,  water.  These  systems  may  have  either 
higher  or  lower  vapor  pressures  than  the  pure  solvents.  The  rela- 
tion of  these  vapor  pressures  to  those  of  the  monovariant  systems, 
two  liquid  phases  and  vapor  is  made  clear  by  following  the  change 
of  pressure  with  the  concentration  at  constant  temperature.  In  the 
concentration-pressure  diagram.  Fig.  18,  are  four  isothermal  curves 
representing  the  behavior  of  typical  pairs  of  partially  miscible  liquids.2 


FIG.    i 8. 

AB  shows  the  pressures  of  the  divariant  system,  solution  and  vapor, 
with  the  less  volatile  component  as  solvent.  At  B  the  solution  be- 
comes saturated  in  respect  to  the  second  component.  BC  is  the 
vapor  pressure  of  the  monovariant  system  with  two  liquid  phases. 
It  is  a  horizontal  line  because  the  pressure  remains  constant  at  con- 
stant temperature  so  long  as  the  two  liquid  phases  are  present,  in- 


1  Walker,  Zeit  phys.  Chem.  5,  192  (1890). 
2Cf.  Konowalow,  Wied.  Ann.  14,  219,  (1881). 


Two  Components  97 

dependent  of  the  percentage  composition  of  the  mixture.  In  pass- 
ing from  B  to  C  the  concentrations  of  the  two  liquid  phases  remain 
constant  for  the  same  reason  that  the  pressures  do  and  the  only 
difference  is  in  the  relative  quantities  of  these  two  phases.  At  B 
there  is  practically  none  of  the  solution  in  which  the  second  com- 
ponent is  solvent ;  at  C  none  of  the  solution  in  which  the  first  com- 
ponent is  solvent.  Since  one  of  the  liquid  phases  disappears  at  C, 
a  further  increase  in  the  relative  amount  of  the  second  compon- 
ent leads  to  the  formation  of  unsaturated  solutions  and  the  pressure 
curve  CD,  for  these  di variant  systems  ends  at  D  with  the  vapor 
pressure  of  the  second  component.  The  characteristic  feature  of 
this  curve  is  that  the  vapor  pressure  of  the  monovariant  system,  rep- 
resented by  BC,  is  higher  than  that  of  either  of  the  single  components. 
This  curve  is  typical  of  most  pairs  of  partly  miscible  liquids,  such  as 
mixtures  of  water  with  chloroform,  toluene,  benzene,  ether  or  naph- 
thalene and  it  was  thought  at  one  time  that  probably  no  other  form 
of  curve  occurred.  The  points  B  and  C  which  represent  the  pres- 
sures and  concentrations  of  the  two  saturated  solutions  wTill  move 
nearer  each  other  as  the  liquids  are  more  soluble  one  in  the  other. 
For  the  hypothetical  case  that  the  t\vo  liquids  are  absolutely  non- 
miscible  the  two  side  curves  AB  and  CD  will  disappear  and  the  line 
BC  will  represent  the  pressures  of  all  the  mixtures.  Under  these 
circumstances  the  vapor  pressure  of  the  system  will  be  the  sum  of 
the  vapor  pressures  of  the  pure  components.  This  case  is  realized 
very  nearly  by  the  system,  chloroform  and  water.1  As  the  mutual 
solubilities  of  the  two  liquids  increase,  the  vapor  pressure  of  the 
monovariant  system  falls  farther  and  farther  below7  the  sum  of  the 
pressures  of  the  two  pure  components.  If  the  vapor  pressure  of  one 
of  the  components  is  low  in  comparison  with  that  of  the  other,  it 
may  readily  happen  that  the  decrease  in  the  vapor  pressure  of  the 
solvent  when  the  less  volatile  component  is  solute  maj^  be  larger  than 
the  partial  pressure  of  the  solute.  This  is  the  normal  case  with 
solutes  which  are  solids  in  the  pure  state  at  the  temperature  of  the 
experiment  and  may  be  expected  with  all  liquid  solutes  wyheu  the 
ratio  of  the  vapor  pressures  of  the  two  components  falls  below  a  cer- 


Ostwald,  Lehrbuch  I,  641. 

7 


98  The  Phase  R^lle 

tain  value  which  is  a  function  of  the  reacting  weights  of  the  pure 
components.3  The  isothermal  vapor  pressure  curve  for  such  a  sys- 
tem is  represented  by  FGHK,  Fig.  18.  FG  and  HK  are  the  curves 
for  the  two  unsaturated  solutions  while  the  two  liquids  and  vapor 
exist  along  GH.  The  difference  between  this  case  and  the  more 
usual  one  is  that  the  vapor  pressure  of  the  monovariant  system  has  a 
value  between  those  of  the  two  components.  The  first  example  of 
such  an  equilibrium  was  observed  by  Roozeboom  *  with  sulfur  diox- 
ide and  water  ;  other  instances  are  sulfur  and  toluene,  sulfur  and 
xylene.3  Now  that  the  conditions  have  been  clearly  defined  under 
which  such  a  system  occurs  there  will  be  no  difficulty  in  increasing 
the  number  of  illustrations  at  pleasure.  If  a  solid  melt  under  a 
liquid  with  formation  of  a  second  liquid  phase,  there  will  be,  near 
the  temperature  at  which  this  takes  place,  a  solution  saturated  with 
respect  to  a  solid  and  having  a  higher  vapor  pressure  than  either 
pure  component  or  two  liquid  phases  with  an  intermediate  vapor 
pressure.  In  other  words,  of  two  cases,  both  of  which  have  been 
considered  abnormal  and  even  impossible,  one  always  occurs. 

A  third  possibility  in  the  way  of  two  liquid  phases  and  vapor  is 
that  the  vapor  pressure  of  this  system  shall  be  less  than  that  of 
either  component.  This  would  be  represented  by  a  curve  of  the 
same  general  form  as  L,MNO.  No  example  illustrating  this  has 
been  found  * ;  but  an  approximation  to  it  has  been  found  in  the  sys- 
tem, hydrobromic  acid  and  water,  which  is  represented  in  the  dia- 
gram by  PQRST.  PQR  is  the  curve  for  the  unsaturated  solution  of 
hydrobromic  acid  in  water.  At  small  concentrations  the  solution 
has  a  lower  vapor  pressure  than  either  component  ;  but  with  increas- 
ing concentration  the  vapor  pressure  of  the  divariant  system  passes 
through  a  minimum,  represented  by  Q,  and  increases  rapidly  to  the 
point  R  where  the  second  liquid  phase  appears  with  and  the  system 
passes  along  RS.  ST  is  the  curve  for  the  solution  of  water  in 
hydrobromic  acid.  I  have  been  unable  to  find  any  data  in  regard  to 


1  Ostwald,  lyehrbuch  I,  643. 

2  Recueil  Trav.  Pays-Bas  3,  31,  (1884)  ;  Zeit.  phys.  Chem.  8,  526,  (1891). 

3  Bancroft,  Jour.  Phys.  Chem.  I,  No.  3(1896)  ;  Hay-wood,  Ibid.  I,  No.  4  (1897). 
4Konowalow  considers  it  unrealizable.     Wied.  Ann.  14,  221,  (1881). 


Two  Components  99 

the  direction  of  this  curve  ;  but  it  probably  ascends  if  one  ma>-  judge 
from  the  behavior  of  sulfur  dioxide  and  water. 

Since  the  two  liquid  phases  in  equilibrium  have  very  different 
compositions,  it  may  be  asked  what  is  the  relation  between  the 
vapor  pressure  of  each  of  the  liquid  phases  and  of  the  system. 
Konowalow  *  showed  experimentally  and  theoretically  that  the  two 
phases  must  give  off  the  same  vapor  and  that  this  was  the  vapor  in 
equilibrium  with  the  two  solution  phases.  Ostwald1  reached 
the  same  conclusion  in  much  the  same  way.  I  quote  his 
reasoning  :  "  Suppose  that  in  a  hollow  ring  (Fig.  19)  A  is  a  satur- 


FIG.    19. 

ated  solution  of  water  in  ether,  B  of  ether  in  water,  and  C  is  the 
vapor  space.  If  the  vapor  in  equilibrium  with  A  had  a  different 
pressure  or  composition  from  the  vapor  in  equilibrium  with  B,  there 
would  be  a  continuous  distillation  or  diffusion  from  one  side  to  the 
other  and  equilibrium  would  never  be  reached  since  the  original 
condition  would  always  be  restored  by  diffusion  through  the  surface 
between  the  solutions.  We  should  thus  have  a  perpetual  motion 
machine,  which  is  impossible."  This  argument  is  not  quite  sound.1 
If  the  partial  pressure  of  each  solute  be  higher  than  the  partial  pres- 
sure of  the  same  component  in  the  solution  in  which  it  is  solvent, 
Ostwald' s  description  of  what  will  take  place  is  accurate.  The  dis- 
tillation from  a  place  of  high  pressure  to  a  place  of  low  pressure  will 
produce  unsaturated  solutions  which  cannot  exist  in  contact  and  this 
assumption  is,  therefore,  untenable.  The  case  is  quite  different  if 


1  Wied.  Ann.  14,  220,  224,  (1881). 
'Lehrbuch,  I,  644. 

3  Bancroft,  Phys.  Rev.  3,  415,  (1896). 


ioo  The  Phase  Rule 

the  partial  pressure  of  each  component  be  assumed  to  be  higher  in 
the  vapor  above  the  solution  in  which  it  is  solvent  than  in  the  vapor 
above  the  solution  in  which  it  is  solute  or,  to  take  a  specific  case,  if 
there  be  more  ether  in  the  vapor  given  off  by  A  than  in  the  vapor 
given  off  by  B,  and  more  water  vapor  given  off  by  B  than  by  A. 
Ether  will  distill  from  A  to  B  and  water  from  B  to  A.  The  water 
condensing  at  A  will  sink  through  the  less  dense  ether  layer  ;  the 
ether  which  condenses  at  B  will  become  saturated  with  water  form- 
ing an  infinitely  thin  film  of  a  solution  of  water  in  ether  on  top  of 
the  aqueous  solution  B.  Both  free  surfaces  having  the  same  com- 
position, there  is  no  possibility  of  further  distillation  if  the  influence 
due  to  gravity  be  neglected.  It  has  been  pointed  out  to  me  by  Pro- 
fessor Trevor  that  the  two  solutions  of  water  in  ether,  not  being  at 
the  same  level,  will  have  different  vapor  pressures  owing  to  the  in- 
fluence of  gravity  and  that  a  final  equilibrium  will  not  be  reached 
until  the  system  has  become  symmetrical.  This  experiment  has 
been  tried  in  his  laboratory  and  the  prediction  verified. 

It  is  thus  possible  for  two  liquid  phases  to  be  in  equilibrium  with- 
out assuming  that  each  sends  off.  the  same  vapor.  It  is  possible  that 
both  the  pressures  and  the  compositions  of  the  two  vapors  may 
differ,  and  consequently  that  the  vapor  in  equilibrium  with  the  two 
liquid  phases  may  differ  both  in  pressure  and  composition  from  the 
vapor  in  equilibrium  with  either  of  the  saturated  solutions.  Under 
these  circumstances  the  horizontal  part  of  the  isothermal  curve  would 
not  meet  the  two  side  curves  but  would  be  connected  with  them  by 
two  practically  vertical  lines  representing  the  state  of  things,  while 
the  second  solution  phase  is  represented  in  such  small  quantity  as 
not  to  have  the  properties  of  matter  in  mass.1  Whether  such  cases 
actually  occur  is  open  to  donbt ;  but  they  are  not  theoretically  im- 
possible on  the  ground  of  violating  the  law  of  the  conservation  of 
energy.  If  one  admits  the  possibility  of  two  liquids  being  absolutely 
non-miscible,  that  would  at  once  furnish  such  a  case  and  might  be  ex- 
emplified by  benzene  and  mercury  for  instance.  If  one  assumes  that 
all  substances,  liquid  or  solid,  are  miscible  to  a  certain  extent,2  the 


1  Gibbs,  Trans.  Conn.  Acad.  3,  129  (1876). 
"Nernst,  Theor.  Chem.  393. 


Components 


question  can  only  be  settled  by  careful  measurements.  If  the  vapor 
given  off  by  the  two  phases  are  identical,  this  is  a  proof  that  the 
change  of  partial  pressure  is  not  the  same  function  of  the  concentra- 
tion for  the  solvent  and  the  solute.1 

Instead  of  measuring  pressures  at  constant  temperature,  it  is  easier 
to  measure  temperatures  at  the  constant  atmospheric  pressure  and  to 
plot  the  change  of  the  boiling  point  with  the  concentration.  This 
will  give  a  curve  of  the  same  general  form  as  the  pressure-concentra- 
tion curve,  but  reversed,  because  a  low  vapor  pressure  implies  a  high 
boiling  point  and  vice  versa.  The  two  forms  of  boiling  point  curves 
for  partially  miscible  liquids  which  have  been  realized  experimentally 
are  given  in  Fig.  20.  They  refer  to  the  same  pairs  of  liquids  as  the 
similarly  lettered  curves  in  Fig.  18.  Thus  ABCD  is  the  typical  boil- 
ing point  curve  for  most  pairs  of  liquids.1  As  will  be  seen  from  the 
diagram,  the  monovariant  system,  two  liquid  phases  and  vapor  has  a 


\ 

K 


FIG.  20. 

lower  boiling  point  than  either  of  the  pure  components.  When  the 
monovariant  system,  two  liquid  phases  and  vapor,  has  a  vapor  pres- 
sure lying  between  those  of  the  single  components,  the  boiling  point 
of  the  system  will  lie  between  the  boiling  points  of  the  pure  sub- 
stances. The  only  instances  of  two  liquids  boiling  in  an  open  flask 
at  temperatures  between  the  boiling  points  of  the  components  are  sul- 
fur and  toluene,  sulfur  and  xylene.  The  boiling  point  curves  for 
these  systems  will  have  the  general  forms  represented  by  FGHK. 


1  Bancroft,  Phys.  Rev.  3,  203  (1895). 

*Cf.  Konowalow,  Wied.  Ann.  14,  142  (1881). 


102  The  Phase  Rule 

While  these  two  types  show  the  same  general  behavior,  a  constant 
boiling  point,  so  long  as  the  monovariant  system  is  present,  they  be- 
have very  differently  when  it  conies  to  the  divariant  system  left  be- 
hind. With  two  liquids  of  the  first  type  as,  for  instance,  isobutyl 
alcohol  and  water,  either  of  the  liquid  phases  may  disappear  on  dis- 
tillation, depending  on  the  relative  quantity  of  the  two  phases.  The 
single  liquid  phase  left  in  the  flask  may  therefore  be  a  solution  of  iso- 
butyl alcohol  in  water,  or  a  solution  of  water  in  isobutyl  alcohol. 
With  sulfur  and  xylene,  on  the  other  hand,  it  will  always  be  the 
phase,  solution  of  sulfur  in  xylene,  which  disappears  on  distillation, 
and  the  liquid  in  the  flask  will  always  be  a  solution  of  xylene  in  sul- 
fur, entirely  irrespective  of  the  relative  amounts  of  the  two  liquid 
phases  in  the  original  mixture.  This  difference  in  behavior  is  shown 
in  the  diagram  ;  both  side  curves  ascend  in  systems  of  the  first  type, 
only  one  in  liquids  of  the  second  type.  Only  a  mixture  with  a  higher 
boiling  point  can  be  left  behind  in  the  flask,  because  the  temperature 
never  sinks  during  a  distillation.1  Since  all  solutions  of  sulfur  in 
xylene  boil  at  a  lower  temperature  than  the  monovariant  system,  they 
can  never  occur  as  residues. 

In  the  diagram  for  naphthalene  and  water,  Fig.  17,  the  curves 
BC  and  ED  have  only  been  followed  a  little  way  ;  but  it  will  be  prof- 
itable to  consider  what  will  be  the  effect  of  increasing  temperature  on 
a  system  composed  of  two  liquid  phases  and  vapor.  The  direction 
of  the  curves  for  the  concentrations  of  the  two  phases  can  be  foretold 
from  the  Theorem  of  L,e  Chatelier.  If  the  solute  dissolves  with  ab- 
sorption of  heat,  its  concentration  will  increase  with  rising  tempera- 
ture, and  vice  versa.  While  it  is  distinctly  exceptional  for  a  solid  to 
dissolve  with  absorption  of  heat  at  ordinary  temperatures,  this  is  by  no 
means  uncommon  in  the  case  of  a  liquid  solute.  There  are  three 
possibilities  in  a  system  composed  of  two  liquid  phases  and  vapor. 
With  rising  temperature  there  may  be  increasing  solubility  in  both 
phases,  or  increasing  solubility  in  one  phase  and  decreasing  solubili- 
ty in  the  other,  or  decreasing  solubility  in  both  phases.  These  three 
cases  have  been. realized  experimentally  by  Alexejew.?  Phenol  and 


An  empirical  generalization. 
:Wied.  Ann.  28,  305  (1886). 


Two  Components  103 

water  is  an  example  of  the  first  case,  the  solubility  of  each  in  the 
other  increasing  with  increasing  temperature.1  Sulfur  and  toluene 
is  another  instance.  The  solubility  of  water  in  isobutyl  alcohol  in- 
creases with  the  temperature,  that  of  isobutyl  alcohol  in  water  de- 
creases at  all  temperatures  below  65°.  Other  examples  are  mixtures 
of  water  with  ether*  or  esters.3  In  all  these  instances  it  is  the  solu- 
bility in  water  which  decreases  with  increasing  temperature.  Be- 
tween the  temperatures  of  o°  and  65°.  both  the  solubility  of  water  in 
secondary  butyl  alcohol  and  that  of  secondary  butyl  alcohol  in  water 
decreases  with  increasing  temperature,  A  yet  more  striking  exam- 
ple of  this  class  is  to  be  found  in  mixtures  of  diethylamine  and  water, 
investigated  by  Guthrie.*  Alexejew  found  that  on  increasing  the 
temperature  sufficiently  all  pairs  of  liquids  finally  come  under  the 
first  heading,  with  solubilities  increasing  with  increasing  tempera- 
ture. Under  these  circumstances  the  compositions  of  the  two  liquid 
phases  will  approach  each  other  until  at  some  temperature  they  be- 
come equal,  and  there  is  only  one  liquid  phase.  Above  this  temper- 
ature the  liquids  are  miscible  in  all  proportions,  or  "consolute." 
The  temperature  at  which  this  takes  place  is  known  as  the  consolute 
temperature.*  The  same  result  has  been  reached  in  a  different  way 
by  Masson,*  starting  from  the  experimental  fact  that  gases  are  misci- 
ble in  all  proportions.  From  this  it  follows  that  two  liquids  must 
become  consolute  at  the  critical  temperature  of  the  mixture,  and  it  is 
probable  that  many  pairs  will  do  so  at  a  much  lower  temperature. 
In  Fig.  2 1  are  the  concentration-temperature  curves  for  some  of  the 
systems  studied  by  Alexejew,  showing  the  different  typical  cases. 
There  is  added  also  the  curve  for  diethylamine  and  water,  because 
the  increase  of  solubility  in  the  two  phases  with  decreasing  tempera- 
ture is  so  great  that  the  liquids  are  consolute  below  a  given  tempera- 


1  If  the  assumption  is  made  that  water  is  the  solvent  in  both  liquid  layers, 
the  concentration  of  phenol  in  the  phase  containing  an  excess  of  that  substance 
should  increase  with  rising  temperature,  which  is  not  the  case. 

*  Nerast,  Theor.  Chem.  390.     *  Bancroft,  Phys.  Rev.  3,  132  ( 1895). 

*Phil.  Mag.  (5),  18,  500  ((886). 

5  Bancroft,  Jour.  Phys.  Chem.  I,  No.  3  ( 1896). 

«Zeit.  phys.  Chem.  7,  500  (1891). 


104 


The  Phase  Rule 


ture.1  Unless  decomposition  occurred  it  would  be  possible  to  find  a 
second  temperature  above  which  the  two  liquids  would  be  miscible 
in  all  proportions. 


FIG.  21. 

In  the  system  made  up  of  naphthalene  and  water,  the  consolute 
temperature  is  so  far  above  the  melting  point  of  naphthalene  that  it 
does  not  appear  on  the  diagram,  Fig.  17,  and  the  two  liquid  phases 
can  be  in  equilibrium  over  a  wide  range  of  temperatures.  With  phe- 
nol and  water  this  is  no  longer  the  case,  and  this  concentration-tem- 
perature diagram  for  this  equilibrium  is  represented  in  Fig.  22.  AB 


1  The  same  phenomenon  occurs  with  ethylamine  and  with  triethylamiue. 


Two  Components 


ic-5 


and  HF  are  the  fusion  curves  for  phenol  and  ice  respectively.1  MBL, 
and  XDL,  are  the  solubility  curves  for  the  two  liquid  phases,  water 
in  phenol  and  phenol  in  water,  the  portions  BM  DX  representing 
states  of  labile  equilibrium  unstable  in  respect  to  solid  phenol.  L,  is 
theconsolute  temperature,  67°-68°,  and  DFisthe  solubility  curve  for 
solid  phenol.  Sometimes  it  occurs  that  the  consolute  temperature  is 


9 
L 


^-<-.-.... 


Ptm*. 


FIG.  22. 

lower  than  the  melting  point  of  the  less  fusible  component,  as  in  the 
case  of  benzoic   acid*  and  water,  Fig.  22,  while  with  salicylic  acid 


1  The  letter  H  has  been  omitted  to  avoid  crowding.  It  belongs  at  the  point 
marked  o°.  For  the  letter  M  in  the  lower  left-hand  square  of  the  diagram, 
read  N. 

1  To  avoid  confusion  this  curve  has  been  reversed,  one  hundred  units  of 
water  being  at  the  right.  The  wavy  form  of  the  curve  AB  is  probably  due  to 
experimental  error. 


io6 


The  Phase  Rule 


and  water,  Fig.  22,  the  whole  curve  ML,N  is  instable.1  Here  AB  is 
unquestionably  a  fusion  curve,  DF  equally  unquestionably  a  solu- 
bility curve,  and  at  some  point  on  the  curve  ABDF  there  must  there- 
fore be  a  change  from  one  to  the  other  ;  but  at  present  we  are  not 
able  to  determine  that  point  exactly,  nor  to  say  anything  in  regard 
to  the  curve  BD  if  B  and  D  do  not  coincide.  It  is  very  much  to  be 
desired  that  these  measurements  of  Alexejew  should  be  repeated  with 
great  care,  as  it  seems  not  impossible  that  the  curve  BD  is  really  a 
straight  line  parallel  to  the  temperature  axis,  and  that  there  is  a  sud- 
den change  of  direction  both  at  B  and  at  D.  A  still  more  remarka- 
ble series  of  curves  is  furnished  by  triethylamine  and  water  in  Fig. 
23.  The  mixtures  of  the  two  liquids  become  turbid  on  warming, 


FIG.  23. 

separating  into  two  liquid  layers.  The  curve  XYZL,  represents  the 
temperatures  at  which  this  takes  place  for  solutions  containing  from 
five  to  ninety-eight  per  cent,  of  water  in  grams.  In  the  diagram  one 
hundred  and  one  grams  of  triethylamine  are  taken  as  equivalent  to 
eighteen  grams  of  water.  No  points  on  the  curve  were  determined 
beyond  L,  but  the  statement  that  a  solution  containing  one  per  cent. 


1  Roozeboom  has  observed  a  similar  phenomenon  with  water  and  a  salt  of 
trinitrophenylmethylnitramine.     Recueil  Trav.  Pays-Bas,  8,  263  (1889). 


Two  Components  107 

of  water  did  not  become  turbid  even  at  200°,  shows  that  the  curve 
LjB  must  represent  the  behavior  of  solutions  containing  very  little 
water.  The  portion  of  the  curve  YZ  was  believed  by  Gnthrie  to  be 
actually  horizontal ;  but  what  this  signifies  is  entirely  unknown.1 

In  the  lion  variant  system,  solid,  two  solutions  and  vapor,  the 
solid  phase  is  not  always  one  of  the  pure  components  ;  but  may  be  a 
compound.  A  most  excellent  example  of  this  form  of  equilibrium 
has  been  found  by  Roozeboom1  in  the  system,  composed  of  sulfur 
dioxide  and  water.  The  solid  phases  which  occur  are  ice  and  the 
hydrate  SO^Hp.  The  symbols  H,O«-rSO,  and  SOs==vH,O  de- 
note a  solution  of  x  reacting  weights  of  SO,  in  one  reacting  weight 
of  water,  and  a  solution  of  y  reacting  weights  of  water  in  one  of 
SO,,  water  being  solvent  in  the  first  case  and  sulfur  dioxide  in  the 
second.  The  pressure- temperature  diagram  for  this  system  is  shown 
in  Fig.  24.  At  B  there  exists  the  nonvariant  system,  ice,  hydrate, 
H5Osss-vSO,  and  vapor.  The  temperature  is  — 2.6°  and  the  pres- 
sure 2i.i'cm.  of  mercury.  BF,  BC,  BZ  and  BL  are  the  boundary 
curves  for  ice,  H,O==:jrSO,  and  vapor  ;  ice,  hydrate  and  vapor  ;  ice, 
hydrate  and  H,Oss=s.rSO, ;  hydrate,  H,Os5==-rSO,  and  vapor  respec- 
tively, while  BA  is  the  labile  prolongation  of  LB.  instable  with  re- 
spect to  ice.  Passing  along  the  curve  BL.  there  is  formed  at  L 
a  new  nonvariant  system  composed  of  hydrate,  H,Oss=-ii>O,, 
SO,  =s=rj-H,O  and  vapor.  The  temperature  at  which  this  system  cau 
exist  is  12.1°  and  the  pressure  177.3  cm-  °f  mercury.  The  curves 
LE.  LX  and  LD  represent  the  monovariant  systems,  H,O=s=rSO,, 
SOjSssjHjO  and  vapor ;  hydrate,  H^Osos-rSO,  and  SO,==:j'H,O  ; 
hydrate,  SO,  ==711,0  and  vapor.  There  is  no  nonvariant  system 
possible  at  the  point  where  the  curve  LD  cuts  BZ  because  the  two 
curves  have  only  one  phase  in  common,  the  hydrate  SO,7H,O.*  LD 
will  terminate  at  some  low  temperature  owing  to  the  formation  of 
solid  sulfur  dioxide,  forming  the  nonvariant  system,  SO,,  SO,7H,O, 
and  vapor ;  but  no  attempt  has  yet  been  made  to  realize 


1  Pickering  has  made  a  careful  study  of  the  freezing  points  of  amines  and 
water.     Jour.  Chem.  Soc.  63,  141  (1893). 

1  Recueil  Trav.  Pays-Has,  3,  29  1 1884)  ;  ZeiL  phys.  Chem.  a,  450  (1888). 
3  Private  letter  from  Professor  Roozeboom. 


io8 


The  Phase  Rule 


this  equilibrium.  The  curve  EL,,  if  continued,  will  lie  below  L,D, 
theoretically  ; l  but  the  difference  in  direction  falls  within  the  limits 
of  experimental  error  and,  therefore,  does  not  show  in  the  diagram. 
Since  the  systems  in  equilibrium  with  solid  sulfur  dioxide  were 
not  investigated,  there  is  but  one  fusion  curve,  FB,  the  other  curves 


200  en 


/80 


SO 


/to 


60 


40 


to 


FIG.  24. 


Nernst's  diagram  is  faulty  in  this  point.     Theor.  Chera.  488. 


Tsro  Components  109 

for  systems  containing  liquid  phases  being  solubility  carves.  Along 
LB  water  is  solvent  and  along  LD  sulfur  dioxide.  Along  LX  and 
LE  sulfur  dioxide  is  solvent  in  one  of  the  two  liquid  phases  and  so- 
lute in  the  other.  It  win  be  seen  that  the  equilibrium  between  sul- 
fur dioxide  and  water  differs  in  one  respect  from  anything  that  has 
yet  been  considered.  The  solvent  is  not  the  same  in  the  two  solu- 
tions which  may  exist  in  equilibrium  with  the  solid  hydrate  SQjH,O, 
while  water  is  solvent  in  both  the  solutions  saturated  with  respect  to 
Fe,CL.i2H,O,  to  take  merely  one  instance.  It  might  seem  at  first  as 
if  this  difference  were  connected  with  the  facts  that  the  solid  hydrate. 
SO.jHjO.  does  not  have  a  true  melting  point,  and  that  it  can  exist 
in  equilibrium  with  two  liquid  phases  simultaneously  ;  but  neither  of 
these  things  is  essential.  These  are  no  theoretical  reasons  known 
why  the  whole  of  the  curve  LE  should  not  represent  a  labile  equi- 
librium, as  was  found  to  be  the  case  experimentally  with  salicylic 
acid  and  water.  Under  these  circumstances,  the  hydrate  would  have 
a  true  melting  point,  and  we  could  not  say  with  our  present  knowl- 
edge whether  the  solvent  changed  or  not.  This  point  can  not  be 
settled  definitely  in  all  cases  until  we  are  able  to  say  at  what  concen- 
tration a  mixture  of  two  consolnte  liquids  changes  from  a  solution  of 
one  in  the  other  to  a  solution  of  the  second  in  the  first. 

In  order  to  define  the  fields  in  which  the  divariant  systems  can 
exist,  it  will*  be  necessary  to  add,  in  Fig.  24,  parts  of  the  pressure- 
temperature  diagrams  for  each  of  the  pure  components.  The  dotted 
VKQR.PP  represents  the  equilibrium  between  liquid  and  gaseous  sulfur 
dioxide ;  YF  between  ice  and  vapor  ;  FY,  between  water  and  water 
vapor ;  FY,  between  ice  and  water.  Ice  and  hydrate  exist  in  the 
field  ZBC  ;  ice  and  HjO^-TSO,  in  ZBFY, ;  ice  and  vapor  in  CBFY ; 
hydrate  and  HjO—jrSO,  in  ZBLX  ;  hydrate  and  vapor  in  CBLD  ; 
Hp—.rSO,  and  vapor  in  Y.FBLE ;  hydrate  and  SO,— j-H,O  in 
XLD  ;  H,O~*SOa  and  SOJ=:j'HaO  in  XT,F.,  while  SO.sfej-H^O 
can  be  in  stable  equilibrium  with  vapor  only  in  the  space  between  the 
lines  DLE  and  pp.*  Increase  of  pressure  causes  any  system  repre- 
sented by  a  point  on  LB  to  pass  into  hydrate  and  vapor,  if  the  quan- 
tity of  vapor  in  the  monovariant  system  be  sufficiently  large  relative- 

Recrcfl  T«T.  Pays-Has,  6,  319  (1887). 


The  Phase  Ride 


ly  to  the  solution  ;  into  hydrate  and  solution  of  sulfur  dioxide  in 
water  if  the  contrary  is  true.  A  further  increase  of  pressure  pro- 
duces the  monovariant  system  represented  by  L,D  on  the  first  assump- 
tion, while,  in  the  second  case,  the  same  change  will  form  the  mono- 
variant  system,  hydrate  and  two  liquid  phases,  if  accompanied  by  a 
rise  of  temperature.  If  the  monovariant  system,  hydrate,  water  in 
sulfur  dioxide,  and  vapor,  be  compressed,  either  the  hydrate  or  the 
vapor  will  disappear,  depending  on  the  relative  quantities  of  the  two. 
Starting  from  any  point  on  L,E,  the  two  phases  which  decrease  with 
increasing  pressure  are  the  solution  of  sulfur  dioxide  in  water  and 
the  vapor  ;  again,  it  is  a  question  of  relative  amounts  whether  one  or 
the  other  of  the  two  possible  di variant  systems  be  formed.  This  will 
be  clear  if  it  be  kept  in  mind  that  the  vapor  contains  both  sulfur  diox- 
ide and  water,  and  that  the  ratio  of  sulfur  dioxide  to  water  in 
the  vapor  is  greater  than  the  ratio  of  the  two  components  in  the 
solid  hydrate  or  in  the  solution  of  sulfur  dioxide  in  water  and 
less  than  the  ratio  in  the  solution  of  water  in  sulfur  dioxide.  As  the 
concentrations  of  the  various  solutions  were  not  all  determined,  it  is 
impossible  to  make  the  concentration-temperature  diagram  for  this 
particular  case  ;  but  in  Fig.  25  is  given  the  general  form  for  two 


FIG.  25. 

components  which  form  one  compound  and  two  liquid  phases.  AB 
is  the  fusion  curve  for  SO2  in  presence  of  water.  At  B  the  hydrate 
crystallizes  out  forming  the  nonvariant  system,  solid  SO2,  SO27H2O, 
solution  of  water  in  SO2,  and  vapor.  Along  BC  we  have  the  solu- 


Two  Components  HI 

bility  curve  of  the  hydrate,  SO,  being  the  solvent.  CL  and  DL  are 
the  two  liquid  phases  and  L  the  consolute  temperature.  SO,  being 
solvent  along  CL  and  water  along  DL-  At  the  temperature  repre- 
sented by  C  and  D  there  exists  the  nonvariant  system,  hydrate,  two 
solutions  and  vapor.  DF  is  the  solubility  curve  for  the  hydrate, 
water  being  solvent,  while  HF  is  the  fusion  curve  for  ke  in  presence 
of  SOr  At  F  there  exists  the  nonvariant  system,  ice,  hydrate,  solu- 
tion of  SO,  in  water,  and  vapor.  This  is  on  the  assumption  that  the 
critical  temperature  of  sulfur  dioxide  is  higher  than  that  of  the  hy- 
pothetical point  L.  Tables  XIX-XX  give  Roozeboom's  data  for 
sulfur  dioxide  and  water,  sixty-four  grams  of  the  former  being  equiv- 
alent to  eighteen  of  the  latter.  The  pressures  are  in  centimeters  of 
mercury  except  for  the  system,  hydrate  and  two  liquid  phases,  where 
the  values  are  given  in  atmospheres. 

TABLE  XTX 


Xonvariant  Systems 


Temp.   ;  Pressure 


Ice,  hydrate  H,O  ==0.02480,,  vapor  —  2.6: 

Hydrate  H.O  =»  o.oSjSC^,  SOt«~jH,O,  vapor        12.1 

TABU*  XX 


21.15 
J77-3 


Temp.         Pressure        Temp.        Pressure      Temp.        Pressure 


Ice, 

hydrate 

Hydrate,  H,O 

«=jcSOt         H^O^-rSOj, 

vapor 

vapor                  SOj^^flU 

J  vapor 

-  9-° 

15.0 

—  6.° 

13-7 

12.1° 

177-3 

—  8. 

16.0 

—  4- 

17-65 

13.0 

:^-5 

—  6. 

17.7 

-    :- 

20.  i 

Hydrate.  H,O=JcSO, 

—  4- 

19-35 

-  2.6 

21.15 

SO, 

=:  • 

H,0 

—  3- 

20.65 

—    2. 

23.0 

I2.I 

:  :  : 

—    2.6 

21.15 

—     I. 

26.2 

I2.9 

20. 

Hydrate, 

i                O.O 

29-7 

13-6 

40. 

vapor 

2.8 

43-2 

14.2 

60. 

+    0.1° 

113-1 

4-45 

14-8 

80. 

3-05 

127.0 

8.00 

66.6 

15-3 

100. 

6.05 

141.9 

8.40 

92.6 

J5-8 

125. 

9-05 

158-2 

IO.OO 

117.7 

16.2 

150. 

H.O 

170.1 

11.30 

150-3 

16.5 

175- 

II.9 

-7-  - 

"-75 

166.6 

16.8 

200. 

12.  1 

177-3 

12.  IO 

177-3 

17-1 

225. 

The  Phase  Rule 


So  far  as  is  known,  sulfur  dioxide  and  water  form  but  one  com- 
pound, SO2yH2O,  and  this  does  not  have  a  true  melting  point.1  Hy- 
drobromic  acid  and  water  form  four  solid  compounds,  HBr4.H2O, 
HBr3H2O,  HBr2H2O  and  HBrH2O,  all  except  the  last  having  a  true 
melting  point.  Fig.  26  is  the  pressure-temperature  diagram  for  part 
of  this  system,3  drawn  to  scale.  The  pressures  are  given  in  atmos- 


FIG.  26. 


1  Roozeboom  has  also  studied  the  equilibrium  between  chlorine  and  water. 
Recueil  Trav.  Pays-Bas,  3,  59  (1884);  4,  69   (1885);   between  bromine  and 
water.     Ibid.  3,  73  (1884)  ;  4,  71  (1885). 

2  Roozeboom,  Ibid.  4,  108,  331  (1884)  ;  5,  323,  351,  353  (1885). 


Two  Components  113 

pheres.  OA  is  the  fusion  curve  for  ice  in  presence  of  hydrobromic 
acid.  The  line  is  dotted  because  this  equilibrium  was  not  studied 
experimentally  by  Roozeboom  and  dees  not  appear  in  his  diagram.1 
Along  AFB  the  dihydrate  exists  in  equilibrium  with  solution  and 
.vapor.  At  F  the  solution  has  the  same  composition  as  the  hydrate. 
The  temperature  of  this  melting  point  is  —  11.30  and  the  pressure 
52.5  cm.  of  mercury.  Along  FB  the  hydrate  is  in  equilibrium  with 
a  solution  containing  more  hydrobromic  acid  than  itself.  Water  is 
still  the  solvent  so  this  is  analogous  to  the  hydrates  of  ferric 
chlorides.  At  B  there  exists  the  non variant  system,  HBr2HsO. 
HBr.H/X  H^Osaa-vHBr  and  vapor,  the  temperature  being  —  15.5° 
and  the  pressure  two  and  one-half  atmospheres.  BC  represents  the 
conditions  under  which  the  dihydrate  and  monohydrate  can  be  in 
equilibrium  with  vapor  while  BL  is  the  curve  for  monohydrate, 
solution  and  vapor.  The  co-ordinates  of  the  point  L  are  — 3.3°  and 
ten  and  one-half  atmospheres.  The  nonvariant  system  existing 
under  these  conditions  is  HBrHsO,  H,Ossss.t  HBr.  HBr«sfH2O  and 
vapor.  The  curves  radiating  from  L  have  the  same  lettering  and 
significance  as  those  in  Fig.  24  if  one  substitutes  hydrobromic  acid 
for  sulfur  dioxide  and  remembers  that  the  solid  phase  has  the  com- 
position HBrH^O.  The  solution  along  LB  contains  less  hydro- 
bromic acid  than  the  crystals  and  if  cooled  at  constant  volume  will 
solidify  completely  at  B  with  formation  of  monohydrate,  dihydrate 
and  vapor.  Roozeboom  performed  no  experiments  below  — 30°,  so 
that  he  studied  only  the  behavior  of  the  two  compounds,  HBr2H,O 
and  HBrHjO.  Pickering*  has  shown  that  there  is  a  compound 
HBr3H,O  with  a  melting  point  at  —  48°  and  a  compound  HBr4H,O 
with  a  melting  point  at  —55.8°.  In  Fig.  26  the  lines  OA  and  BFA 
are  drawn  and  lettered  as  if  they  came  together  at  A.  This  is  ob- 
viously inaccurate  but  the  lines  could  not  be  kept  separate  unless  a 
very  different  scale  were  used.  In  reality  the  line  BFA  terminates 
at  about  — 48°  with  the  appearance  of  the  trihydrate  and  the  forma- 
tion of  the  mono  variant  s\-stem,  HB^H/D,  solution  and  vapor.  At 
about  — 57°  there  can  exist  the  nonvariant  sj-stem,  HBr4H,O, 


1  Zeit.  phys.  Chem.  2,  454  (1888). 
!Phfl.  Mag.  (5)36,  in 


u4  Tke  Pkase  Rule 

HBr3H2O,  solution  and  vapor.  The  tetrahydrate  melts  at  — 55.8°. 
The  dotted  curve  OA  has  been  followed  as  far  as  — 73°  and  the  pre- 
cipitate was  still  ice.  It  is  not  known  at  what  temperature  the  next 
nonvariant  system  occurs  nor  whether  the  new  phase  is  tetrahydrate 
or  some  other  compound.  The  systems,  hydriodic  acid  and  water, 
hydrochloric  acid  and  water,  are  very  similar  to  the  one  just  dis- 
cussed and  bring  out  no  new  points.1  In  tables  XXI-XXIII  are 
Roozeboom's  data  for  hydrobromic  acid  and -water,  x^  denotes  units 
of  hydrobromic  acid  per  unit  of  water  ;  x^  units  of  water  per  unit  of 
hydrobromic  acid  ;  xs  units  of  hydrobromic  acid  in  one  hundred 
units  of  solution.  Eighty-one  grams  of  hydrobromic  acid  are  equiv- 
alent to  eight  grams  of  water. 

TABLE   XXI 


Nonvariant  Systems 


Temp.       Pressure 


HBr2H2O,  HBrH,O,  H2O=^o.6iHBr,  vapor        —15.5°      195  cm. 
HBrH,O,  H,O~o.83HBr,  HBr=^jH2O,  vapor    —3.3        io>^Atni. 
Melting  point  of  HBraH.,0  —  1 1.3       525  cm. 


TABLE!  XXII 


Temp. 

Pressure 

*, 

x.2      x^     Temp. 

Pressure     x^ 

*, 

-, 

HBr2H2O,  solution  and  vapor.         HBr2H2O,  solution  and  vapor 

—  25.°  !      o.i  cm.  0.39 

2.56 

28.0—11.5°        lAtm.    0.5211.9134.3 

—  21.8  <          T. 

—  12.0          i#        0.54.1.8535.1 

-18.9         3. 

0.42 

2.38 

29.6—12.6          i^        0.551.81135.6 

-16.8 

6. 

-13-3          i# 

—  14.6 

12. 

0.45 

2.24 

3O.8   —  14.0              2                  0.59    I.yO  37.0 

-13.0 

22. 

—  14.8              2% 

—  12.4 

28. 

—  15.5 

2^/z            O.6l 

1.63 

38.0 

—  12.0 

34- 

0.48 

12.  IO 

32.3 

—  ii.  6 

44- 

—  11.3       52.5         0.50 

J2.OO 

33-3 

'Roozeboom,  Recueil  Trav.  Pays-Bas,  3,  84  (1884);    Zeit.  phys.  Cliem.  2, 
459  (1888)  ;  Pkkering,  Phil.  Mag.  (5)  36,  rn 


Two  Components 


Temp. 


-14.8 


HBrH,O,  solution  and  vapor. 
0.61  1.64 


37-9 


—  14.0 

3 

0.62 

.61 

38-3 

—  II.O 

4 

0.65 

-55 

39-3 

-  8.7 

5 

0.67 

-50 

40.0 

-  7-2 

6 

0.69 

-45 

40.9 

-  5-8 

7 

0.72 

.38 

42.0 

-  4-7 

8 

0.76 

-32 

43-i 

-  4-0 

9 

-  3-3 

0.83           i 

[.20 

45-5 

TABUB  XXIII 

Temp.   Pressure  Temp.  Pressure   Temp.  Pressure  Temp.  Pressure 

HBr2H,O,  HBrH,O  and  rapor  HBrELO,  H.O  s=a:.rHBr,  HBr=ss:j-HjO 

—28.5°;     76cm.  —20°      131  cm.  —3.3°  lo^Atm  — 1.6°  looAtm. 

—26.          85          — 18        156  —2.9    25  —0.9     150 

—24.     *     96          — 16        184  —2.4    50  —0.3    200 

—22.        in          —15-5    195  —2.0    75  +0.3    250 


CHAPTER  VIII 


CONSOLUTE  LIQUIDS 

Since  a  fusion  curve  ends  at  its  intersection  with  the  solubility 
curve,  it  is  clear  that  the  greater  the  solubility  the  farther  the  fusion 
curve  can  be  followed  and  that  this  may  even  be  carried  to  such  an 
extent  that  the  fusion  curve  for  one  component  may  meet  the  fusion 
curve  for  the  other  component.  This  actually  occurs  in  a  great 
many  instances  and  has  often  been  looked  upon  as  the  typical  case 
from  which  all  equilibria  could  be  derived  by  adding  the  necessary 
limitations.  It  is  more  rational  to  treat  it  as  a  subhead  of  the  nor- 
mal case,  exemplified  in  Figs.  17,  22,  in  which  the  solubility  curves 
have  disappeared.  As  an  example  we  will  take  the  equilibrium  be- 
tween naphthalene  and  phenanthrene  studied  by  Miolati,1  Fig.  27. 

1  Zeit.  pbys.  Chem.  9,  649  (1892). 
1  Phil.  Mag.  (5)  17,  462  (1884). 


FIG.  27. 

Addition  of  phenanthrene  lowers  the  freezing  point  of  naphthalene 
and  the  temperatures  and  concentrations  at  which  the  monovariant 
system,  naphthalene,  solution  and  vapor,  can  exist  are  shown  by  the 
curve  AB.  Starting  with  phenanthrene  and  adding  naphthalene 
there  is  a  lowering  of  the  freezing  point  and  the  curve  CB  represents 
the  conditions  of  equilibrium  for  the  system,  phenanthrene,  solution 
and  vapor.  At  B,  48°,  there  exists  the  nonvariant  system,  naph- 
thalene, phenanthrene,  solution  and  vapor.  At  this  point  the  solu- 


Ttf-o  Components  117 

tion  solidifies  without  change  of  temperature.  The  mixture  with 
the  lowest  freezing  point  was  called  by  Guthrie  the  eutectic  alloy 
and  he  looked  upon  the  ciyohydric  mixture  as  a  special  case  of  the 
same  thing.  There  is  one  important  difference  between  the  two 
cases.  Salt  and  ice  appear  at  the  intersection  of  a  solubility  and  a 
fusion  curve,  naphthalene  and  pheuanthrene  at  the  intersection  of 
two  fusion  curves.  It  is  desirable  to  distinguish  between  these  two 
cases  in  the  following  way.  The  temperature  at  which  a  binary 
solution  in  equilibrium  with  vapor  passes  completely  into  two  solid 
phases  is  called  the  ciyohydric  temperature,  if  a  solubility  aud  a 
fusion  curve  meet  at  this  point  and  the  eutectic  temperature  if  the 
intersection  is  between  two  fusion  curves.  This  does  not  include 
the  third  case  in  which  a  solution  can  solidify  without  change  of 
temperature  when  two  solubility  curves  meet,  and  the  solution  has  a 
composition  between  those  of  the  two  solid  phases.  This  is  realized 
when  FelCl,i2HaO,  Fe.Cl.jH/),  solution  and  vapor  are  in  equilib- 
rium, but  there  seems  to  be  no  need  of  a  special  name  for  this  tem- 
perature. 

In  the  field  ABC  there  exists  the  divariant  system,  solution  and 
vapor.  At  all  temperatures  higher  than  the  melting  point  of  the  less 
fusible  component,  there  can  be  no  solid  phase  if  there  are  no  com- 
pounds possible,  and  the  solution  will  be  a  mixture  of  two  consolute 
liquids.  If  the  two  components  form  compounds  or  solid  solutions, 
it  is  merely  necessary  to  raise  the  temperature  above  the  melting 
point  of  the  least  fusible  solid  phase  in  order  to  reach  the  same  state 
of  things.  The  divariant  system,  solution  and  vapor,  is  also  made 
up  of  two  consolute  liquids  when  the  temperature  of  the  experiment 
is  higher  than  that  at  which  two  liquid  phases  can  coexist  with  vapor. 
Strictly  speaking,  there  are  two  classes  of  consolute  liquids,  those 
which  never  form  two  liquid  phases  under  any  circumstances  and 
those  which  sometimes  do.  Our  knowledge  of  the  subject  is,  unfor- 
tunately, so  imperfect  that  it  is  impossible  from  a  study  of  the  vapor 
pressures  of  mixtures  of  consolute  liquids  to  predict  what  new  phase 
will  separate  under  given  conditions.  The  different  ways  in  which 
the  pressure  changes  with  the  concentration  at  constant  temperature 
are  shown  in  the  concentration-pressure  diagram,  Fig.  28.  Four 
styles  of  curves  have  been  found  experimentally,  having  a  maxi- 


n8 


The  Phase  Rule 


mum,  a  minimum,  both  a  maximum  and  a  minimum  and  neither  a 
maximum  nor  minimum  value.  In  systems  represented  by  AA,  ad- 
dition of  either  component  to  the  other  produces  an  increase  in  the 
vapor  pressure,  and  there  will  therefore  be  some  concentration  for 
which  the  vapor  pressure  has  a  maximum  value  higher  than  that  of 
either  of  the  pure  components.  Instances  of  this  are  to  be  found  in 
mixtures  of  propyl  alcohol '  or  butyric  acid  with  water  ;  in  mixtures 
of  carbon  bisulfide  with  ethyl  alcohol  or  ethyl  acetate  ; 2  in  carbon 
tetrachloride  and  methyl  alcohol.3  In  systems  represented  by  BB, 


FIG.  28. 

addition  of  either  compound  to  the  other  causes  a  lowering  of  the 
vapor  pressure  so  that  there  will  be  some  concentration  of  the  liquid 
phase  in  equilibrium  with  a  vapor  having  a  minimum  pressure  lower 
than  that  of  either  of  the  components.  Mixtures  of  water  with 
formic,  nitric,  and  the  haloid  acids  come  under  this  head.4  There 
are  no  instances  known  of  an  isothermal  pressure  curve  having  a 
maximum  and  a  minimum  value,  the  one  being  higher  and  the  other 
lower  than  that  of  either  component,  though  this  is  a  perfectly  con- 
ceivable case  and  might,  pehaps,  be  realized  if  some  one  were  to  study 
mixtures  of  one  of  the  haloid  acids  and  water  at  a  suitable  tempera- 
ture. There  are  two  instances  known  where  the  isothermal  pressure 


1  Konowalow,  Wied.  Ann.  14,  50  (1881). 

2  Brown,  Jour.  Chem.  Soc.  39,  529  (1881). 
:t  Thorpe,  Ibid.  35,  544  (1879). 
4Konowalow,  Wied.  Ann.  14,  51   (1881). 


Two  Components  119 

curve  has  the  form  represented  by  CC,  with  both  a  maximum  and  a 
minimum  value,  lying  between  those  for  the  pure  components. 
These  two  cases  are  propionic  acid  and  water1  at  64°,  benzene  and 
carbon  tetrachloride*  at  34.8°.  It  is  by  no  means  certain  that  this 
form  of  curve  really  exists.  Vapor  pressure  measurements  are  not 
easy  to  make  accurately,  and  it  may  well  ba  that  the  wavy  nature  of 
the  curve  is  due  to  experimental  error.  With  propionic  acid  and 
water  this  is  the  more  probable  conclusion,  since  the  difference  be- 
tween the  maximum  and  the  minimum  is  small  and  the  data  for  con- 
structing the  curve  were  obtained  by  interpolation  and  not  by  direct 
measurement.  With  benzene  and  carbon  tetrachloride  the  matter  is 
a  little  different.  It  is  necessary  to  assume  an  error  of  some  two 
centimeters  in  a  total  of  seventeen  in  order  to  account  for  the  phe- 
nomenon. This  means  an  error  of  at  least  ten  per  cent.,  and  would 
detract  greatly  from  the  importance  of  Linebarger's  work  if  estab- 
lished. It  is  much  to  be  regretted  that  the  unusual  nature  of  the 
results  was  not  recognized  and  especial  pains  taken  to  verify  or  dis- 
prove them.  The  fourth  class  of  curve  is  that  represented  by  DD, 
having  neither  maximum  nor  minimum.  The  curve  may  be  either 
convex  or  concave,  and  is  typical  of  most  pairs  of  consolute  liquids. 
It  is  not  known  what  is  the  relation  of  this  curve  to  the  others.  If 
two  consolute  liquids  have  the  same  vapor  pressure  at  a  given  tem- 
perature, some  mixture  of  the  two  must  have  either  a  maximum  or 
a  minimum  vapor  pressure  at  that  temperature  unless  all  concentra- 
tions have  the  same  value.  If  at  other  temperatures  the  isothermal 
curve  has  neither  maximum  nor  minimum  it  is  possible  to  pass  by 
change  of  temperature  from  a  system  behaving  in  one  way  to  a  sys- 
tem behaving  in  another.  This  would  prevent  any  deductions  from 
the  form  of  the  pressure  curve  at  one  temperature  to  that  at  another. 
The  alternative  is  that  all  consolute  liquids  with  intersecting  vapor 
pressure  curves*  form  solutions  with  a  maximum  or  minimum  value 
at  some  concentration.  No  experimental  work  on  this  subject  has 
been  done. 


1  Konowalow,  /.  c.  45. 

'Linebarger,  Jour.  Am.  Chem.  Soc.  17,  690  (1895). 

;IGuye,  ZeiL  phys.  Chem.  14,  570  (1894). 


i2o  The  Phase  Rule 

In  order  to  treat  the  subject  of  fractional  distillation,  it  is  neces- 
sary to  consider  the  boiling  points  of  mixtures  of  consolute  liquids. 
In  the  concentration-temperature  diagram,  Fig.  29,  are  represented 
the  three  important  types  of  boiling  point  curves.  The  systems 
which  have  a  maximum  vapor  pressure  have  a  minimum  boiling 
point  and  vice  versa,  while  systems  with  the  vapor  pressures  of  all 
mixtures  lying  between  the  values  for  the  pure  components  will 
have  intermediate  boiling  points.  Although  the  quantitative  data 
in  regard  to  the  composition  of  the  vapor  in  equilibrium  with  a  given 
solution  are  few  and  far  between,  there  is  little  difficulty  in  deter- 
mining them  qualitatively,  and  these  compositions  are  given  schemati- 
cally by  the  dotted  lines  in  the  diagram. 


FIG.    29. 

The  composition  of  the  vapor  at  any  temperature  is  given  by  the 
point  at  which  the  horizontal  line  for  that  temperature  meets  the 
dotted  curve.  At  the  temperature  minimum  and  maximum,  A:  and 
Bt,  the  vapor  has  the  same  percentage  composition  as  the  liquid  and 
these  solutions  will  therefore  distill  without  change  of  temperature. 
This  has  been  confirmed  experimentally,  the  mixture  of  propyl  alco- 
hol and  water  with  the  lowest  boiling  point  and  that  of  formic  acid 
and  water  which  has  the  highest  boiling  point,  behaving  like  pure 
liquids.  All  other  mixtures  in  so  far  as  they  are  represented  in  the 
diagram  have  the  two  components  present  in  different  proportions  in 
the  vapor  and  the  liquid  phases.  For  convenience  it  will  be  assumed 
that  the  left  side  of  the  diagram  represents  one  hundred  per  cent. 


Tav  Components  121 

water  and  the  right  side  one  hundred  per  cent,  of  the  other  compo- 
nent. In  the  equilibrium  between  propyl  alcohol  and  water  repre- 
sented by  AAtA  all  the  solutions  to  the  left  of  A,  contain  more  water 
than  the  vapor,  while  all  to  the  right  contain  more  propyl  alcohol 
than  the  vapor.  Partial  distillation  yields  a  distillate  richer  in  pro- 
pyl alcohol  than  the  residue  in  the  first  case  and  one  richer  in  water 
in  the  second  case.  By  continued  or  fractional  distillation  of  a  mix- 
ture of  propyl  alcohol  and  water,  there  is  obtained  in  the  distillate 
the  mixture  with  the  lowest  boiling  point  and  composition  At  which 
cannot  be  purified  further  in  this  way.  In  the  distilling  flask  there 
is  left  behind  pure  water  if  the  original  solution  contained  more 
water  than  the  mixture  with  constant  boiling  point  and  pure  propyl 
alcohol  if  the  contrary  was  the  case.  With  liquids  like  formic  acid 
and  water.  BB,B.  the  phenomena  are  reversed.  Solutions  to  the  left 
of  Bj  contain  less  water  than  the  vapor  ;  solutions  to  the  right  less 
formic  acid.  By  fractional  distillation  the  first  set  give  pure  water, 
the  second  pure  formic  acid  in  the  distillate,  while  both  leave  behind 
in  the  flask  the  solution  of  formic  acid  and  water  with  the  highest 
boiling  point.  This  cannot  be  separated  into  its  components  by 
further  fractioning.  because,  as  has  been  pointed  out.  the  solution 
and  vapor  have  the  same  composition.  In  Table  XXIV  are  given 

TABLE  XXIV 


H,0 

HC1 

100° 

: 

110° 

20.2 

H.O 

HBr 

TOO 

-64 

116 

47-8 

ILO 

HI 

IOO 

—34 

127 

57 

H,O 

HCOOH 

IOO 

+99-9 

107.1 

:.  c 

H.O 

HXOs 

IOO 

86 

120.5 

68.0 

HC1 

(CH,)S0 

-  83 

—  21 

—       2 

61 

H,0 

C,H7OH 

IOO 

+974 

+  97  4 

77 

CC1, 

CH3OH 

76.6 

65-2 

65-2 

21.9 

the  compositions  and  boiling  points  of  some  mixtures  corresponding 
to  the  points  At  and  B,.  In  the  first  and  second  columns  are  the  form- 
ulas representing  the  two  components ;  in  the  third  and  fourth  columns 
the  boiling  points  of  the  first  and  second  components  respectively  ; 
in  the  fifth,  the  boiling  point  of  the  mixture  with  constant  composi- 
tion, and  in  the  sixth  the  composition  of  that  mixture  expressed  in 


122  The  Phase  Rule 

grams  of  the  second  component  in  one  hundred  grams  of  the  solu- 
tion. Strictly  speaking  hydrobromic  acid  and  water  should  not  be 
considered  as  a  mixture  of  two  consolute  liquids  under  atmospheric 
pressure  because  the  solid  hydrate  HBr2H2O  separates  when  the 
concentration  of  the  hydrobromic  acid  passes  a  certain  limit.  It 
might  be  urged  that  all  the  haloid  acids  are  gases  in  the  pure  state  at 
ordinary  temperatures  under  atmospheric  pressure  and  this  point  is 
a  sound  one.  On  the  other  hand  it  has  been  deemed  wise  to  include 
them  in  the  table  because  they  are  very  familiar  examples  of  solu- 
tions with  constant  boiling  points  higher  than  those  of  either  of  the 
pure  components.  These  solutions  with  constant  boiling  points  have 
received  a  great  deal  of  attention  and  the  attempt  was  early, made  to 
treat  them  as  compounds.  Roscoe1  showed  that  this  could  not  be 
the  case  because  the  mixtures  did  not  conform  to  the  Theorem  of 
Definite  and  Multiple  Proportions  and  because  the  composition  of 
the  mixture  with  a  constant  boiling  point  was  a  function  of  the  pres- 
sure under  which  the  boiling  took  place.  The  mixture  of  hydro- 
chloric acid  and  water  which  distills  without  change  of  temparature 
contains  eighteen  per  cent,  of  hydrochloric  acid  when  the  external 
pressure  is  180  cm.  of  mercury  and  twenty-three  and  two-tenths  per 
cent,  when  it  equals  5  cm.  Hydrobromic  and  hydriodic  acid  behave 
in  the  same  way,  the  solution  containing  more  acid  at  lower  tempera- 
tures and  pressures.  Nitric,  hydrofluoric  and  formic  acids  show  the 
opposite  behavior,  the  amount  of  acid  in  the  mixture  with  constant 
boiling  point  increasing  with  the  external  pressure.  We  are  not 
able  to  predict  the  direction  of  this  change.  It  is  evident  from  the 
curves  AAtA,  BB,B  that,  with  systems  having  a  minimum  or  maxi- 
mum boiling  point,  there  can  be  two  solutions  with  the  same  boiling 
point  at  any  temperature  between  the  minimum  boiling  point  and 
the  boiling  point  of  the  more  volatile  component  in  the  one  case  and 
between  the  maximum  boiling  point  and  the  boiling  point  of  the 
more  volatile  component  in  the  other  case.  A  similar  relation  will 
hold  in  regard  to  the  pressures  at  constant  temperature.  It  has  been 
pointed  out  by  Gibbs*  that  the  temperatures  for  which  there  can  be 

'Liebig's  Annalen,  l'l2,  327  (1859);  1*6,  203  (1860)  ;  Cf.  Ostwald,  L,ehr- 
buch,  I,  649. 

2  Trans.  Conn.  Acad.  3,  156  (1876). 


Two  CenHpomfmts 


123 


two  solutions  with  the  same  pressure  and  the  pressures  for  which 
there  can  be  two  solutions  with  the  same  temperature  can  be  repre- 
sented graphically.  In  the  pressure-temperature  diagram.  Fig-  30, 
there  is  a  qualitative  reproduction  of  the  behavior  of  propyl  alcohol 
and  water.  The  pressure-concentration  curve  BB,B  of  Fig.  28  be- 
comes a  straight  line  ;  B,  is  the  vapor  pressure  of  pure  water  at  that 
temperature.  B  of  pure  propyl  alcohol  and  B,  of  the  solution  having 
the  same  composition  as  the  vapor.  Suppose  this  to  be  done  for 
several  temperatures  and  curves  drawn  through  all  the  B"s,  all  the 
B,"s  and  all  the  B,"s.  In  the  field  below  the  B,  line  there  can  exist 
only  vapor.  In  the  field  between  the  B,  line  and  the  B  line  occur  all 
the  simultaneous  values  of  temperature  and  pressure  for  which  only 
one  pair  of  coexistent  phases  is  possible.  Between  the  B  line  and 
the  B,  line  are  all  the  simultaneous  values  of  temperature  and  pres- 
sure for  which  there  are  two  pairs  of  coexistent  phases,  while  above 
the  last  line  there  can  exist  a  liquid  phase  only. 


FIG.  30. 

In  the  diagram.  Fig.  29,  the  dotted  line  DD,D  is  typical  of  all 
systems  with  intermediate  boiling  points  which  have  yet  been  stud- 
ied. As  the  position  of  the  curve  shows,  the  vapor  always  contains 
a  greater  percentage  of  the  more  volatile  component  than  the  liquid. 
Through  fractional  distillation,  if  carried  on  long  enough,  the  two 
liquids  will  finally  be  separated  completely,  the  more  volatile  being 
found  in  the  distillate  while  the  less  volatile  remains  in  the  boiling 


124  The  Phas?  Rule 

flask.  The  Hempel  colunui  is  merely  an  apparatus  by  which  a  large 
series  of  distillations  is  carried  out  in  what  seems  to  be  a  single  oper- 
ation. The  vapor  is  partially  condensed  on  the  first  set  of  beads, 
the  uncondensed  part  becoming  richer  in  the  more  volatile  compo- 
nent, and  this  process  is  repeated  through  the  whole  length  of  the 
column.  The  condensed  liquid  flowing  down  through  the  tube 
washes  off  the  beads  so  that  the  lower  end  of  the  column  contains  a 
liquid  having  a  relatively  large  amount  of  the  more  volatile  compo- 
nent and  this  takes  yet  more  of  the  less  volatile  component  out  of 
the  vapor.  Other  things  being  equal  the  column  is  more  effective 
when  in  full  action  than  when  the  distillation  is  first  bsgun.  The 
ease  of  separation  depends  on  the  difference  between  the  percentage 
compositions  of  the  vapor  and  the  solution.  If  there  is  a  large  differ- 
ence the  two  liquids  can  be  separated  easily  ;  otherwise  not.  This 
can  not  be  tola  from  the  boiling  point  curve  alone,  and  it  is  not  cor- 
rect to  state,  as  Ostwald  l  has  done,  that  the  ease  of  separation  de- 
pends on  the  pitch  of  the  curve.  It  is  conceivable  that  there  might 
be  found  a  pair  of  consolute  liquids  which  would  give  an  almost 
straight  boiling  curve  and  yet  have  the  composition  of  the  two  phases 
nearly  identical  at  each  temperature.  On  the  other  hand,  it  is  to  be 
noticed  that  when  the  boiling  points  of  the  two  pure  components  lie 
very  near  together,  the  difference  between  the  composition  of  the 
solution  and  vapor  phases  is  apt  to  be  small,  and  that  such  liquids 
can  not  readily  be  separated  by  fractional  distillation. 

It  seems  not  impossible  that  the  curve  for  the  composition  of  the 
vapor  phase  might  have  the  form  DD4D  or  DD3D,  though  no  exam- 
ple of  this  is  known.  Under  these  circumstances  there  would  be  one 
solution,  represented  by  Dt  in  the  one  case  and  by  D3  in  the  other, 
which  would  distill  unchanged  though  the  temperature  is  neither  a 
maximum  nor  a  minimum.  Bauer*  claims  to  have  found  that  a  solu- 
tion containing  equal  reacting  weights  of  propylene  and  ethylene 
bromide  distills  completely  at  136°,  a  temperature  between  the  boil- 
ing points  of  the  two  components.  This  statement  has  found  its  way 
into  Beilstein's  Handbuch  ;3  but  I  am  informed  by  Professor  Orn- 

1  Lehrbuch,  I,  648. 

"2Liebig's  Annalen,  Suppl.  I,  250  (1861). 

:'Vol.  I,  36. 


Tu-o  Components  125 

dorff  that  it  is  not  correct.  A  pair  of  liquids  with  the  vapor  compo- 
sition curve  DD,D  would  distill  so  that  the  mixture  with  constant 
boiling  point  would  remain  in  the  flask,  and  one  or  the  other  of  the 
pure  components  would  be  formed  in  the  distillate  depending  on  the 
original  composition  of  the  solution.  If  the  vapor  curve  were  repre- 
sented by  DD,D  the  mixture  with  the  composition  D,  would  distill 
off,  leaving  one  of  the  pure  components  in  the  flask.  In  both  these 
cases  we  should  have  the  very  unusual  phenomenon  of  the  boiling 
point  falling  with  continued  distillation,  something  which  is  entirely 
outside  of  our  experience.  It  is  very  much  to  be  desired  that  some 
one  should  undertake  a  careful  study  of  the  vapor  pressures,  both 
total  and  partial,  of  mixtures  of  consolute  liquids.  At  present  the 
measurements  of  Liuebarger  are  practically  all  that  we  have  on  the 
subject,  and  the  accuracy  of  these  is  by  no  means  unquestionable.1 
Lehfeldt*  has  made  a  few  measurements  on  the  composition  of  the 
vapors  in  equilibrium  with  different  solutions  at  the  same  tempera- 
ture ;  but  he  has  not  determined  the  vapor  pressure.  There  is  also 
one  set  of  experiments  by  Winkelmanu*  on  mixtures  of  propyl  alco- 
hol and  water.  No  attention  has  been  paid  to  the  very  curious  vapor 
pressure  curves  found  by  Guthrie,4  one  of  which  certainly  deserves 
careful  examination,  alcohol  and  amyleue. 

While  it  is  impossible  to  say  anything  absolute  about  the  behav- 
ior of  two  liquids  when  cooled  below  the  temperature  at  which  they 
are  consolute,  it  is  clear  that  there  is  a  general  connection  between 
the  form  of  the  concentration-pressure  curve  and  the  nature  of  the 
phase  which  separates.  The  monovariant  system,  solid,  solution  and 
vapor,  will  give  a  curve  with  neither  maximum  nor  minimum,  if  the 
system  has  a  lower  vapor  pressure  than  that  of  one  pure  component 
and  higher  than  that  of  the  other.  This  is  the  usual  case.  The 
curve  will  have  a  maximum  if  the  monovariant  system  has  a  higher 
vapor  pressure  than  either  pure  component.  If  the  monovariant 
system,  two  liquid  phases  and  vapor,  has  a  higher  vapor  pressure  at 
the  consolute  temperature  than  either  of  the  pure  components,  the 


'Jour.  Am.  Chem.  Soc.  17,  615,  690  ( 1895). 
'Phil.  Mag.  (5)  40,  397  (i»9&). 
3  Wied.  Ann.  39,  i  (1890). 
*Phil.  Mag.  (5)  18,  515  (1884)- 


126 


The  Phase  Rule 


pair  of  consolute  liquids  will  give  a  curve  with  a  maximum  pressure. 
This  is  the  usual  case,  though  there  are  no  quantitative  measure- 
ments illustrating  it.  If  the  vapor  pressures  of  the  monovariant  sys- 
tem lie  between  those  of  the  two  components,  the  resulting  divariant 
system  will  have  all  its  pressures  lying  between  those  of  the  two  pure 
components.  As  there  are  no  experiments  on  this  point  it  is  not  pos- 
sible to  state  that  such  a  curve  has  a  change  of  curvature  somewhere, 
though  it  is  very  probable.  A  minimum  vapor  pressure  will  occur 
if  the  solution  on  cooling  forms  the  monovariant  system,. solid,  solu- 
tion and  vapor,  with  a  pressure  less  than  that  of  either  pure  compo- 
nent. This  can  only  happen  when  the  solubility  coefficient  of  each 
vapor  in  the  other  liquid  is  very  large  ;  but  this  is  merely  another 
way  of  saying  the  same  thing  and  does  not  explain  anything,  because 
we  are  not  in  a  position  to  make  predictions  in  regard  to  the  solubil- 
ity coefficients. 

Where  there  are  two  liquid  phases  in  equilibrium,  one  of  the 
components  is  solvent  in  the  one  phase,  the  other  in  the  other.  Each 
component  has  a  definite  and  known  solubility  in  the  the  other.  At 
the  consolute  temperature  the  two  phases  have  the  same  composition 
and  become  miscible  in  all  proportions.  At  this  temperature  there 


/Y 


FIG.  31. 

are  two  consolute  liquids  each  with  a  definite  and  known  solubility 
in  the  other,  the  two  values  being  reciprocal.  This  raises  the  ques- 
tion of  the  form  of  the  solubility  curves  at  yet  higher  temperatures. 
The  solubility  of  either  liquid  in  the  other  may  become  infinite  above 
the  consolute  temperature  or  it  may  not.  The  graphical  representa- 
tion of  these  two  cases  is  given  in  the  concentration-temperature  dia- 
gram, Fig.  31.  AX  A,  and  BXBj  are  the  .solubility  curves  if  the  two 


Tan  Components  127 

solubilities  become  infinite  at  the  consolute  temperature ;  CYC,  and 
DYD,  are  the  solubility  curves  for  a  system  in  which  no  such  change 
takes  place  at  the  consolute  temperature,  here  represented  by  Y.  At 
temperatures  lower  than  Y,  any  mixture  represented  by  a  point  in 
the  field  ACYD  separates  into  two  liquid  phases  whose  concentra- 
tions are  given  by  the  points  on  CY  and  DY  corresponding  to  the 
temperature  of  the  experiment.  Above  the  consolute  temperature  a 
mixture  corresponding  to  any  point  in  the  field  C,YD,  will  not  form 
two  liquid  layers  because  the  mixture  is  unsaturated  in  regard  to 
either  component  as  solvent.  In  this  field  and  in  this  field  only  is  it 
a  matter  of  complete  indifference  which  component  is  taken  as  sol- 
vent. That  such  systems  exist  is  shown  by  the  fact  that  the  curves 
CY  and  D\"  are  exact  representations  of  the  behavior  of  sulfur  and 
toluene.1  There  can  be  little  doubt  that  the  solubilities  represented 
by  these  lines  do  not  become  infinite  at  the  consolute  temperatures. 
We  see  thus  that  it  is  not  necessary  for  two  liquids  to  have  infinite 
solubilities  one  in  the  other  in  order  to  be  miscible  in  all  proportions  : 
but  that  the  same  result  is  attained  if  the  solubilities  overlap.1 

Many  consolute  liquids  have  definite  but  unknown  solubilities, 
one  in  the  other.  The  determination  of  these  unknown  solubilities 
is  one  of  the  problems  yet  to  be  solved.  An  examination  of  the  pres- 
sure-concentration curves  at  constant  temperature  for  mixtures  of  dif- 
ferent alcohols  and  water,  shows  that  there  is  probably  a  relation  be- 
tween the  form  of  the  curve  and  the  solubility.  It  is  well  known 
that  decreasing  amount  of  carbon  in  homologous  series  means  usu- 
ally increasing  solubility  in  water,  and  we  find  that,  as  we  pass  down 
the  series,  we  have  the  isobntyl  alcohol  forming  two  liquid  phases 
with  water,  giving  the  characteristic  curve  AA.  Fig.  32.  Propyl 
alcohol  is  miscible  in  all  proportions  with  water,  and  the  concentra- 
tion-pressure curve  BB  has  a  distinct  maximum ;  with  decreasing 
amount  of  carbon  we  have  ethyl  alcohol  and  water,  represented  by 
CC  ;  methyl  alcohol  and  water,  represented  by  DD.  The  curve  CC 
is  distinctly  convex,  as  seen  from  above,  while  this  is  no  longer  the 
case  with  DD.*  It  should  be  kept  in  mind  that  the  decrease  in  the 


1  Alexejew,  Wk*L  Ann.  *8.  310  ( 1886). 

SC£  Horstmann,  Graham- Otto's  Lehrbuch.  I, 

*06twald.  Lebrbnch  I.  647  ;  Bancroft,  Jour.  Phys.  Chem.  I,  No.  3  « iSg6>. 


128 


The  Phase  Ride 


reacting  weights  of  the  alcohols  tends  to  modify  the  curves  in  the 
same  way,  and  it  is  not  impossible  that  the  effect  of  increasing  but 
unknown  solubility  may  be  very  slight  or  even  non-existent. 

With  salicylic  acid  and  water  it  has  been  seen  that  a  solubility 
curve  may  appear  to  be  a  continuation  of  a  fusion  curve  and  it  is 
therefore  not  safe  to  assume  that  because  the  concentration-tempera- 
ture diagram  for  a  binary  system  has  the  general  form  of  Fig.  27, 
the  curve  AB,  for  instance,  is  necessarily  a  fusion  curve  along  its 
whole  length.  If  it  is  a  single  curve,  it  must  be  a  fusion  curve  be- 
cause it  starts  from  the  melting  point  of  one  of  the  pure  components 


FIG.   32. 

which  no  solubility  curve  can  do.  If,  at  any  point,  there  is  a  sud- 
den change  of  direction,  there  must  be,  at  that  point,  the  change 
from  a  fusion  to  a  solubility  curve  with  the  other  component  as  sol- 
vent if  the  precipitate  remains  unchanged.  When  it  is  remembered 
that  the  change  'of  direction  in  passing  from  one  curve  to  another 
may  be  infinitely  slight  and  that  no  system  has  been  studied  with 
the  object  of  determining  whether  such  discontinuity  occurs,  it  is  not 
surprising  that  there  is  a  deplorable  lack  of  data  in  regard  to  this 
point.  The  experiments  of  Fytard *  are  the  only  ones  which  are 
available.  He  finds  a  distinct  break  for  silver  nitrate  and  water  at  a 
temperature  of  about  45°,  and  for  sodium  nitrate  and  water  at  about 
60°.  Silver  nitrate  occurs  in  two  modifications  ;  but  the  change 


Comptes  rendus,  xo8,  176  (1889). 


Two  Components  129 

from  one  to  the  other  takes  place  at  159.5°  and  does  not  seem  to  be- 
tray itself  in  the  curve  for  the  equilibrium  between  silver  nitrate,  so- 
lution and  vapor.  This  is  undoubted!}-  due  to  the  large  experimental 
error  in  the  neighborhood  of  the  fusion  point.  There  is  also  a  distinct 
break  in  the  curve  for  triphenylmethane  and  carbon  bisulfide1 ;  but 
this  may  bs  due  to  the  appearance  of  another  modification  of  tri- 
phenylmethane since  one  is  known.2  It  is  rather  strange  that  there 
should  be  no  break  when  the  carbon  bisulfide  is  replaced  by  chloro- 
form or  hexane.  Tilden  and  Shenstone3  have  found  in  barium  ace- 
tate and  water  a  mixture  giving  a  curve  which  resembles  that  for  the 
system,  solid  salicylic  acid,  solution  and  vapor.  It  it  evident  that 
there  is  a  change  with  increasing  concentration  of  barium  acetate  from 
a  solubility  to  a  fusion  curve  but  it  is  impossible  to  tell  from  the  ex- 
periments at  what  point  this  takes  place.  In  all  these  cases  the 
point  at  which  the  solution  solidifies  without  change  of  tempera- 
ture is  the  cryohydric  point  and  not  the  eutectic  point.  It  is 
probable  that  in  few  mixtures  of  salts  and  water  is  there  an  intersec- 
tion of  two  fusion  points  ;  but  the  subject  calls  for  a  great  deal  more 
careful  study  than  has  yet  been  devoted  to  it.  The  formation  of  a 
true  eutectic  mixture  by  the  meeting  of  two  fusion  curves  is  not  con- 
fined to  organic  substances,  though  the  bulk  of  the  work  has  been 
done  upon  them  as  involving  fewer  experimental  inconveniences.4 
It  is  probable  that  the  mixtures  of  two  salts  which  do  not  crystallize 
together  are  instances  of  true  eutectic  mixtures,  provided  the  salts 
are  consolute  in  the  liquid  state.  In  Table  25  are  given  the  compo- 
sitions of  some  binary  salt  solutions  which  solidify  without  change  of 
temperature  and  the  freezing  points  of  these  mixtures.5  The  com- 
positions are  expressed  in  reacting  weights  of  the  salt  mentioned  in 
one  hundred  reacting  weights  of  the  solution.  The  other  salt  is,  in 
all  cases,  potassium  nitrate  which  melts  at  320°.' 


1  Comptes  rendus,  115,  950  (1892). 

2  Lehmann,  Molekularphysik,  I,  202. 

3  Phil.  Trans.  175,  23  (1884). 

*Vignon,  Comptes  rendus,  1x3,  133  (1881)  ;  Miolati,  Zeit.  phys.  Chetn.  9, 
649  (1892) ;  Dahms,  Wied.  Ann.  54,  486  (1895) ;  Roloff,  Zeit.  phys.  Chem.  17, 
325(1895). 

5Guthrie,  Phil.  Mag.  (5)  17,  643  (1884). 

6Cf.  Ostwald,  Lehrbuch  I,  1023. 
9 


130 


The  Phase  Rule 
TABLE  XXV 

Temp. 

:    Cone. 

Potassium  chroraate 

295° 

2.0 

Calcium  nitrate 

251 

17-3 

Strontium  nitrate 

~5» 

14-3 

Barium  nitrate 

278 

I4.O 

Lead  nitrate 

207 

21.2 

Potassium  sulfate 

300 

1.4 

Sodium  nitrate 

215 

36.8 

If  a  solution  containing  two  components,  which  can  not  crystal- 
lize together  nor  form  two  liquid  phases,  be  cooled  it  will,  at  length, 
become  saturated  in  respect  to  one  component.  If  there  is  no  super- 
saturation  this  will  precipitate  out  and  the  temperature  will  fall  until 
the  point  is  reached  where  tlfe  other  component  separates  in  the  solid 
form.  From  this  moment  the  temperature  will  remain  unchanged 
until  the  whole  of  the  solution  has  disappeared.  There  will  be  only 
one  point  at  which  the  temperature  remains  constant,  though  the 
rate  of  cooling  will  change  when  the  first  solid  phase  appears.  If, 
however,  the  solution  becomes  supersaturated  and  cools  a  degree  or 
two  below  the  temperature  at  which  the  solid  phase  should  appear, 
there  will  be  a  sudden  rise  of  temperature  when  the  solid  phase  actu- 
ally separates  out.  If  the  mass  of  the  liquid  be  fairly  large  and  the 
thermometer  not  too  sensitive,  it  will  seem  as  if  the  temperature  re- 
mained stationary  for  a  short  time.  The  temperature  at  which  this 
takes  place  will  change  with  the  concentration  of  the  solution  where- 
as the  temperature  at  which  the  solution  finally  disappears  is  inde- 
pendent of  the  original  concentration.  This  phenomenon  of  the  sta- 
tionary and  movable  freezing  point  is  really  a  very  simple  one,  the 
movable  point  being  the  one  that  is  determined  in  all  cryoscopic 
measurements.  In  aqueous  solutions,  it  has  been  familiar  to  every- 
one for  years  ;  but  when  it  was  noticed  in  melted  alloys  it  aroused  a 
great  deal  of  interest  and  was  the  cause  of  many  hypotheses.1 

If  the  components  form  one  or  more  compounds,  the  case  will 
be  different.  If  no  one  of  the  compounds  has  a  true  melting  point, 


'Cf.  Ostwald,  Lehrbuch  I,  1018-1027.     The  first  satisfactory  treatment  of  the 
subject  is  due  to  Schnltz,  Pogg.  Ann.  ±37,  247  (1869). 


Two  Components  x-ji 

there  will  be  only  one  concentration  at  which  the  solution  can  solidi- 
fy without  change  of  temperature  ;  while  for  every  hydrate  with  a 
true  melting  point  there  will  be  two  more  such  concentrations.  In 
Fig.  33  are  typical  curves  for  the  freezing  points  of  binary  systems  in 
which  the  components  can  combine  to  form  compounds  but  not  solid 
solutions. 


FIG.  33- 

The  curve  AAjAjA3A4A  represents  a  system  which  forms  three 
compounds,  no  one  of  which  is  stable  at  its  melting  point.  This 
system  is  realized  very  nearly  in  the  equilibrium  between  potassium 
hydroxide  and  water,1  the  three  compounds  being  KOFfyHjO, 
KOH2H2O  and  KOHH2O.  As  a  matter  of  fact,  the  monohydrate 
reaches  its  melting  point  just  before  the  curve  for  the  anhydrous  salt 
begins ;  but,  ignoring  that,  there  is  only  one  concentration,  repre- 
sented by  Aj,  at  which  the  solution  will  solidify  without  change  of 
temperature,  the  solid  phases  being  ice  and  KOIfyH/X 


1  Pickering,  Jour.  Chem.  See-  63,  890  (1893). 


132 


The  Phase  Rule 


is  a  case  exemplified  by  methylamine  and  water.1  At  B,  the  solution 
solidifies  to  ice  and  the  trihydrate.  B2  is  the  melting  point  of  the 
trihydrate,  and  a  solution  having  that  composition  solidifies  without 
change  of  temperature.  At  B3  the  solid  phases  are  probably  methyl- 
amine and  the  trihydrate,  though  this  has  not  been  determined  ex- 
perimentally. CCjC^C  is  the  curve  for  a  system  forming  two  com- 
pounds, one  of  which  has  a  true  melting  point.  Trimethylamine 
and  water  illustrate  this  case,  the  two  hydrates  having  the  composi- 
tion (CH5)sNnH,Oand  (CHj)sN2H,O,  though  this  latter  compound 
has  not  been  analyzed.  The  solution  will  solidify  without  change  of 
temperature  at  C,  to  ice  and  (CHj)sNuH2O  ;  at  C2  to  the  hydrate 
alone,  and  at  Cs  to  trimethylamine  and  the  dihydrate.  DD,D2D3D4D5D 
is  a  concentration-temperature  diagram  for  a  system  forming  two 
compounds,  each  stable  at  its  fusion  point.  This  case  is  realized  in 
diethylamine  and  water  where  the  solid  hydrates  are(C2H5)3NH  i  iH2O 
and  H2O2(C2H5)2NH.  As  will  be  seen  from  the  diagram,  there  are 
five  concentrations  at  which  the  solution  will  solidify  without  change 
of  temperature.  It  would  be  possible  to  make  diagrams  for  still 
other  hypothetical  cases  and  find  illustrations  for  them,  as  for  in- 
stance, that  there  should  be  three  compounds  possible,  of  which  the 
middle  one  only  should  not  be  stable  at  the  fusion  point.  This 
would  be  represented  by  dimethylamine  or  ethylamine  and  water. 
One  simple  rule  covers  all  cases.  If  the  number  of  compounds  sta- 
ble at  the  melting  point  be  "  n,"  the  number  of  solutions  which  will 
solidify  without  change  of  temperature  is  "  2n  -+-  i,"  if  only  stable 
states  of  equilibrium  are  considered.  It  follows  that  there  can  never 
be  an  even  number  of  solutions  which  will  do  this,  a  rule  which  would 
be  very  serviceable  in  case  two  of  the  points  were  very  close  together, 
as  they  well  might  be. 

The  change  of  direction  which  occurs  when  a  fusion  curve  pass- 
es into  a  solubility  curve  is  not  to  be  confounded  with  the  "  break  " 
when  a  second  modification  of  the  solvent  separates.  In  the  latter 
case  there  is  formed  the  nonvariant  system,  two  solid  modifications  of 
the  solvent,  solution  and  vapor,  and  the  temperature  will  remain  con- 
stant so  long  as  the  four  phases  are  present.  In  the  solubility  deter- 


1  Pickering,  Jour.  Chetii.  Soc.  63,  141  (1893). 


Two  Components 


133 


initiations  of  Etard1  there  is  a  very  distinct  point  of  discontinuity  in 
the  curve  for  potassium  nitrate  and  water  at  about  125°.  Guthrie's 
measurements  show  the  same  phenomenon  at  about  130°  though  not 
so  clearly.1  Neither  of  them  paid  any  attention  to  the  nature  of  the 
crystals  at  this  point ;  but  there  is  little  doubt  that  a  second  modifi- 
cation of  potassium  nitrate  crystallizes  out.  Such  a  modification  is 
known  and  its  inversion  temperature  was  found  to  be  129. 5°.*  In 
Fig.  34  is  a  graphical  reproduction  of  Etard's  results  for  silver  and 


Iff       LW       Uv 


/rr/£ 


\ 


K 


'\fff     .+0     \&     \zf      /a 


FIG.  34. 

potassium  nitrates  and  potassium  chlorate.  In  the  first  case  the 
break  is  due  to  the  change  of  solvent,  in  the  second  case  to  a 
change  in  the  nature  of  the  solid  phase  and  in  the  third  case  to  one 
of  the  two.  Potassium  chlorate*  occurs  in  two  modifications  but 
there  are  no  satisfactory  data  as  to  the  inversion  temperature  so  that 


1  Comptes  rendus,  108,  176  (1889). 

'Phil.  Mag.  (5)  x8,  114  (1884). 

3  Schwarz.  Prize  Dissertation,  Gottingeu  ( 1892). 

*  Lehmann,  Molekularphysik,  I,  215. 


134  The  Phase  Rule 

it  is  impossible  to  say  to  what  this  change  of  direction  is  due.  Ac- 
cording to  Ktard  there  is  a  second  break  in  the  curve  for  potassium 
nitrate  ;  but  it  is  a  little  unsafe  to  draw  any  conclusions  from  it  be- 
cause Guthrie's  measurements  show  no  sign  of  it. 

All  the  nonvariant  systems  containing  two  components  which 
have  been  considered  thus  far  have  been  made  up  either  of  two  solid 
phases,  solution  and  vapor  or  of  one  solid  phase,  two  solutions  and 
vapor.  It  is  also  possible  to  have  a  nonvariant  system  containing  no 
liquid  phase  and  composed  of  three  solid  phases  and  vapor.  An  ex- 
ample of  this  is  to  be  found  in  the  double  salt  of  silver  and  mercuric 
iodides,  HgI22AgI.  This  salt  changes  color  at  50°  and  it  has  been 
assumed  that  the  change  was  analogous  to  the  one  taking  place  with 
mercuric  iodide,  the  formation  of  another  modification  of  the  same 
salt.4  Quite  recently  it  has  been  shown  by  Bauer2  that  the  double 
salt  breaks  up  into  its  components  forming  the  nonvariant  system, 
mercuric  iodide,  silver  iodide,  double  salt  and  vapor,  an  equilibrium 
which  can  exist  at  one  temperature  and  one  pressure  only.  In  view 
of  this,  it  is  quite  possible  that  the  change  in  the  double  iodide  of 
copper  and  mercury,  which  takes  place  at  88°,  may  also  be  a  decom- 
position. 


1  Lehman n,  Molekularphysik  I,  169. 
5Zeit.  phys.  Chem.  x8,  1 80  (1895). 


CHAPTER  IX 


SOLID   SOLUTIONS 

There  still  remain  to  be  considered  the  cases  in  which  there  can 
exist  one  or  more  solid  phases  with  concentrations  varying  continu- 
ously within  certain  limits  ;  in  other  words,  the  cases  in  which  it  is 
possible  to  have  formation  of  solid  solutions.  No  system  of  this  class 
has  been  studied  in  detail,  so  that  the  discussion  will  have  to  be  lim- 
ited to  a  statement  of  the  few  facts  already  known  and  a  reference  to 
some  of  the  equilibria  yet  to  be  realized.  We  have  already  seen  that 
the  freezing  point  of  a  solution  is  invariably  lower  than  that  of  the 
pure  solvent,  if  the  latter  crystallizes  in  the  pure  form.  This  is  not 
necessarily  true  if  a  solid  solution  separates.  The  possibilities  are 
best  seen  in  the  pressure-temperature  diagram,  Fig.  35.  AO  is  the 


FIG.  35. 

vapor  pressure  curve  for  the  pure  solvent  as  liquid,  OB  for  the  same 
as  solid,  CC,  is  the  pressure  curve  for  a  solution.  If  the  pure  solvent 
crystallizes  the  freezing  point  of  the  solution  will  be  at  C,,  a  temper- 
ature lower  than  O.  If  a  solid  solution  separate,  the  partial  pressure 
of  the  solvent  in  the  solid  solution  will  be  less  than  the  vapor  pres- 
sure of  the  pure  solvent  at  that  temperature.  The  partial  pressures 
of  the  solvent  in  the  solid  solution  may  be  represented  by  the  curve 


136  The  Phase  Rule 

DD  if  the  solid  solution  is  a  dilute  one,  and  by  EE  if  it  is  a  concen- 
trated one.  The  liquid  and  solid  solutions  will  be  in  equilibrium,  so 
far  as  we  know,  when  the  partial  pressure  of  the  solvent  in  the  vapor 
phase  is  the  same  for  the  two  solutions.1  The  freezing  point  of  a 
given  solution  is  the  temperature  at  which  the  pressure  curve  cuts 
the  pressure  curve  for  the  particular  solid  solution  with  which  it  can 
be  in  equilibrium.  In  the  two  cases  represented  in  the  diagram 
these  freezing  points  are  at  Ct  and  C3.  Both  these  temperatures  are 
higher  than  C1;  while  C3  is  higher  than  O.  The  freezing  point  of  a 
liquid  solution  is  always  higher  if  a  solid  solution  separates  than  if 
the  pure  solvent  crystallizes  ;  if  the  solid  solution  is  sufficiently  con- 
centrated, the  freezing  point  is  higher  than  that  of  the  pure  solvent. 
Both  these  cases  have  been  realized  experimentally.  The  first  case 
is  exemplified  by  thiophene  and  benzene,2  w-cresol  and  phenol,  iodine 
and  benzene,3  and  a  number  of  other  mixtures.*  It  was  to  account 
for  the  abnormally  small  depressions  of  the  freezing  point  that  van  't 
Hoff D  developed  the  conception  of  solid  solutions.  A  rise  of  freezing 
point  has  been  observed  in  many  cases  by  Krister.45  If  the  two  com- 
ponents are  miscible  in  all  proportions  in  the  solid  phase,  a  nonvari- 
ant  system  is  impossible  since  there  can  not  be  two  solid  phases  by 
hypothesis,  and  it  is  improbable  that  two  substances  which  are  con- 
solute  in  the  solid  phase  will  not  be  consolute  in  the  liquid  phase. 
Under  these  circumstances  we  shall  expect  to  find,  in  the  freezing 
points  of  such  systems,  all  the  types  which  were  found  for  the  boil- 
ing points  of  pairs  of  consolute  liquids.  There  will  be  mixtures 
having  a  minimum  freezing  point  lower  than  that  of  either  of  the 
pure  components ;  there  will  be  cases  where  the  freezing  point  of 
some  mixture  is  higher  than  that  of  either  of  the  components,  and, 
lastly,  there  will  be  instances  where  the  freezing  ooints  of  all  mix- 


1  This  leaves  out  of  account  the  partial  pressure  of  the  solute  which  has  not 
received  a  quantitative  treatment.     The  most  plausible  assumption  is  that  the 
partial  pressures  of  the  solute  are  equal  for  the  two  solutions  when  the  system 
is  in  equilibrium. 

2  van  Bijlert,  Zeit.  phys.  Chem.  8,  343  (1891). 
3Beckmann  and  Stock,  Ibid.  17,  123  (1895). 

4Ferratini  and  Garelli,  Ibid.  13,  i  (1894)  ;  Garelli,  18,  51  (1895);  Paterno, 
Ibid.  19,  191  (1896). 

*  Ibid.  5,  322  (1890).     "Ibid.  8,  577  (1891). 


Tu'o  Components  \yj 

tures  lie  between  those  of  the  pure  components.  Only  the  last  type 
has  been  studied  as  yet ;  but  in  this  class  there  has  already  been 
found  a  limiting  case  which  is  entirely  new.  It  is  clear  that  if  a  solid 
solution  has  the  same  composition  as  the  liquid  from  which  it  sepa- 
rates, the  mixture  will  solidify  without  change  of  temperature.  If, 
in  a  given  system,  the  solid  solution  always  has  the  same  composi- 
tion as  the  liquid  phase  from  which  it  crystallizes,  it  follows  that  each 
mixture  of  these  two  components  will  behave  like  a  homogeneous 
body  with  its  own,  definite,  constant  freezing  point.  An  example 
of  this  has  been  found  in  the  system  composed  of  hexachlor- 
rt-keto-y-R-pentane  and  pentachlormonobrom-«  -keto-y-R-pentane. ' 
The  analogous  case  of  two  liquids,  such  that  the  composition  of  the 
vapor  is  always  the  same  as  that  of  the  solution  with  which  it  is  in 
equilibrium,  has  never  been  realized. 

If  the  two  components  are  not  miscible  in  all  proportions  in  the 
solid  phase,  there  are  several  nonvariant  systems  possible.  If  each 
component  is  slightly  soluble  in  the  other  solid  component,  there  can 
exist  the  nouvariant  system,  two  solid  solutions,  liquid  solution  and 
vapor.  An  instance  of  this  seems  to  occur  in  the  system  composed 
of  quinonedihydroparadicarboxylic  ester  and  succinylosuccinic  ester.  - 
Thallium  and  potassium  chlorates*  probably  offer  another  illustration 
of  this  equilibrium,  though  this  has  never  been  shown  experiment- 
ally. If  one  component  is  soluble  in  the  other,  and  the  second  in- 
soluble in  the  first,  there  will  be  possible  the  nonvariant  system, 
solid  solution,  one  pure  component  as  solid  phase,  solution  and  vapor. 
This  has  never  been  observed,  but  it  seems  probable  that  benzene 
and  iodine  come  under  this  head.  The  two  solid  phase's  may  also  be 
a  solid  solution  and  a  compound.  If  the  two  components  form  two 
sets  of  solid  solutions  and  a  compound  or  three  sets  of  solid  solu- 
tions, the  possible  nonvariant  systems  are  increased,  to  say  nothing 
of  allotropic  modifications  or  of  the  possibility  of  two  liquid  phases. 
It  is  a  waste  of  time  to  attempt  to  classify  these  different  equilibria 
until  there  are  some  experimental  data  upon  the  subject,  and  they 
are  referred  to  here,  chiefly,  to  call  attention  to  our  lack  of  knowl- 
edge. 

'Kiister,  Zcit  phys.  Chern.  5,  601  (1890). 
'Lehmami,  Zeit.  phys.  Chem.  I,  49  (1889). 
s  Roozebooui,  Ibid.  8,  531  (1891). 


138  The  Phase  Rule 

It  has  already  been  shown  that  when  only  the  pure  components  or 
compounds  can  separate  from  the  solutions,  the  number  of  mixtures 
for  which  the  solution  solidifies  without  change  of  temperature  is 
2  n  -f  /  where  n  is  the  number  of  compounds  with  stable  melting  points. 
If  a  solid  solution  separates,  this  rule  does  not  apply,  and  all  that 
can  be  said  is  that  if  a  solution  solidifies  without  change  of  tempera- 
ture, and  the  freezing  pcint  is  not  a  maximum,  there  must  have  been 
formed  two  solid  phases,  one  richer  and  one  poorer  in  either  compo- 
nent than  the  solution. 

Passing  to  divariant  systems  we  have  to  consider  the  absorption, 
adsorption  or  occlusion  of  gases  by  solids.  These  three  terms  prob- 
ably refer  to  the  same  phenomenon  in  most  cases  and  yet  the}'  are 
often  used  as  though  distinct.  By  man}'  people  absorption  is  used 
for  the  solution  of  gases  and  vapors  in  liquids,  adsorption  for  the  con- 
densation of  gases  upon  solids, l  while  occlusion  often  carries  with  it 
the  idea  of  a  hypothetical  mechanical  entanglement.  These  distinc- 
tions are  rarely  carried  through  consistently,  and  the  solution  of 
gases  in  metals  is  spoken  of  indifferently  as  absorption  or  occlusion. 
This  general  haziness  in  language  is  the  sign  of  a  corresponding  lack 
of  clearness  in  ideas.  When  a  porous  substance  is  brought  into  con- 
tact with  a  liquid,  phenomena  due  to  capillary  action  will  certainly 
occur,  and  this  may  also  take  place  when  a  gas  is  substituted  for  the 
liquid.  On  the  other  hand,  capillary  action  will  not  account  for  all 
the  phenomena  observed,  and  it  is  rather  doubtful  whether  it  is  an 
important  factor  in  most  cases.  It  would  be  well  to  keep  the  term 
"adsorption"  for  effects  which  may  prove  to  be  due  primarily  to 
surface  tension,  and  to  treat  "  absorption  "  as  the  general  term  ap- 
plying to  liquid  and  solid  solvents,  while  "  occlusion  "  would  refer 
only  to  the  formation  of  solid  solutions.  The  conception  of  mechan- 
ical entanglement  is  to  be  given  up  as  not  describing  the  facts. 

The  solubility  of  gases  in  solids  is  much  greater  than  is  usually 
supposed,  such  different  substances  as  wool,  glass  and  metais  being 
able  to  condense  them.2  Deville3  first  showed  experimentally  that 
hydrogen  diffused  through  many  metals  ;  but  his  method  was  a 

1  Du  Bois-Reyuiond,  cf.  Ostwald,  Lehrbuch  I,  1084. 
*  Cf.  Ostwald,  Lehrbuch  I,  1084-1098. 
3Cf.  Lehmaun,  Molekularphysik  II,  81. 


Tzco 


139 


rough  one  and  yielded  good  results  only  at  high  temperatures.  We 
owe  to  Helmholtz1  a  very  delicate  method  based  on  the  polarization 
of  platinum  by  occluded  hydrogen,  and  Thoma*  has  determined  in 
this  way  which  metals  have  the  power  of  occluding  hydrogen  to  any 
perceptible  extent. 

In  applying  the  Phase  Rule  to  these  phenomena  we  are  practi- 
cally limited  to  the  system,  palladium  and  hydrogen.  Since  this  is 
a  di  variant  system  at  ordinary  temperature  and  pressure,  ft  must  be 
possible  for  the  system  to  have  any  pressure  at  any  temperature,  and 
this  is  the  case  experimentally.  It  also  follows  that  for  a  given  tem- 
perature and  pressure  the  concentration  of  the  occluded  gas  must 
always  have  the  same  value.  This  is  not  true  experimentally.  "  Pal- 
ladium foil  and  palladium  black  absorb  different  amounts  of  hydro- 
gen under  the  same  conditions.  The  explanation  of  this  anomaly  is 
not  hard  to  find,  Solids  are  rarely  homogeneous.  There  is  a  differ- 
ence between  the  surface  and  the  interior,  and  there  are  often  stresses 
throughout  the  mass  due  to  the  methods  of  preparation.  Such  a 
solid  is  not  really  in  equilibrium  ;  but  the  passive  resistances  to 
change  are  so  great,  or  the  reaction  velocity  is  so  low.  that  the  final 
equilibrium  may  never  be  reached.  If  two  pieces  of  metal  are  not 
exactly  alike  there  is  no  reason  that  the  absorption  coefficients  should 
be  identical.  This  may  seem  like  a  far-fetched  explanation  :  but 
any  one,  who  has  compared  two  pieces  of  metal  electrically,  knows 
what  differences  may  and  do  exist  between  different  parts  of  the  same 
rod.  Under  the  circumstances  the  values  for  the  occlusion  of  hydro- 
gen by  palladium  are  not  absolute  because  they  depend  upon  the 
sample  of  metal  used,  or  rather  upon  the  treatment  it  has  received. 
By  always  using  metal  prepared  under  the  same  conditions,  it  is  pos- 
sible to  obtain  comparable  values 

From  a  study  of  the  relation  between  the  pressure  of  the  vapor 
phase  and  the  concentration  in  the  solid  phase,  van  't  Hoff  *  decided 
that  the  compound  PdJH  was  first  formed,  and  that  the  hydrogen 
occluded  in  excess  of  this  formed  a  solid  solution  of  hydrogen  in  pal- 


'Gcs.  Abh.  I.  835. 

:  Zcit.  phjs.  Chan.  3,  69  (1889). 

Tt::   5.  —      --•: 


140  The  Phase  Ride 

ladium  hydride.  The  careful  measurements  of  Robzeboom  and  Hoit- 
sema1  have  failed  to  confirm  this  view.  They  find  that  the  pressure- 
concentration  diagram  is  composed  of  three  parts.  With  increasing 
concentration  of  hydrogen  in  the  solid  phase  the  pressure  increases 
up  to  a  certain  value.  Beyond  this  point  the  pressure  remains  near- 
ly constant  while  the  concentration  in  the  solid  phases  increases. 
When  this  latter  has  reached  a  certain  value,  the  pressure  and  con- 
centration, vary  again  simultaneously.  This  form  of  curve  was  en- 
tirely unexpected,  and  it  is  not  certain  how  it  is  to  be  explained. 
Hoitsema  is  in  doubt  whether  this  points  to  a  condensation  of  hydro- 
gen to  the  liquid  form,  or  to  the  formation  of  two  solid  solutions. 
Neither  of  these  explanations  is  very  satisfactory  and  the  matter  must 
be  left  open  for  the  present,  the  more  especially  since  the  latest  work 
on  the  occlusion  of  hydrogen  by  platinum2  has  not  yet  led  to  any 
definite  conclusion  in  regard  to  the  nature  of  the  solid  phase  or  phases 
present. 

In  the  case  of  the  occlusion  of  gases  by  carbon  there  is  little 
doubt  that  we  have  the  formation  of  a  solid  solution.  The  experi- 
ments of  Chappuis*  bring  out  the  striking  analogy  between  the  ab- 
sorption of  gases  by  liquids  and  by  solids.  The  different  kinds  of 
charcoal  showed  very  different  absorption  powers  ;  but  this  is  not 
surprising  when  one  considers  the  variations  in  chemical  properties 
under  the  same  circumstances.* 


'Zeit.  phys.  Chetn.  xy,  i  (1895). 

*  Mond,  Ramsay  and  Shields,  Ibid.  19,  25  ( 1896). 

*Wied.  Ann.  12,  161  (1881). 

*Cf.  Meslans,  Etats  allotropiques,  112. 


CHAPTER  X 

REVIEW 

Before  passing  to  systems  made  up  of  three  components,  it  will 
be  well  to  give  a  brief  summary  of  the  different  kinds  of  nonvariant 
systems  studied,  with  an  illustration  of  each,  so  far  as  this  is  possi- 
ble. In  the  nonvariant  systems  made  up  of  two  solid  phases,  solu- 
tion and  vapor,  the  two  solid  phases  are  the  two  components  as  sol- 
vents in  the  case  of  naphthalene  and  pheuanthrene  ;  the  two  compo- 
nents, one  as  solvent,  the  other  as  solute,  in  the  case  of  ice  and  po- 
tassium chloride,  ice  and  naphthalene  ;  solute  and  compound  in  the 
case  of  sodium  snlfate  and  hydrated  sodium  sulfate  ;  two  compounds 
in  the  case  of  the  ferric  chlorides  with  twelve  and  seven  of  water  or 
iodine  monochloride  and  trichloride ;  two  modifications  of  the  sol- 
vent in  the  system,  potassium  nitrate  and  water ;  two  modifications 
of  the  solute  in  the  system,  sulfur  and  toluene  ;  two  solid  solutions 
in  the  case  of  quinonedihydroparadicarboxylic  ester  and  succinylo- 
succinic  ester,  or  possibly  potassium  and  thallium  chlorates.  The 
cases  where  the  two  solid  phases  are  a  compound  and  solid  solution 
or  solvent  and  solid  solution,  have  not  been  realized  with  certainty, 
while  the  case  of  solute  and  solid  solution  probably  occurs  with  iodine 
and  benzene,  though  this  lacks  experimental  confirmation. 

There  have  been  two  kinds  of  nonvariant  systems  studied,  in 
which  there  are  two  solutions,  solid  and  vapor.  In  one  the  solid 
phase  is  one  of  the  components,  a  case  realized  by  water  and  naph- 
thalene ;  in  the  other  the  solid  phase  is  a  compound,  exemplified  in 
the  equilibrium  between  sulfur  dioxide  and  water.  In  the  nonvari- 
ant system,  three  solid  phases  and  vapor,  there  can  be  the  two  com- 
ponents and  a  compound  as  solid  phases.  This  has  been  discovered 
in  the  system  composed  of  silver  iodide  and  mercuric  iodide.  There 
are  no  cases  known  where  a  solid  solution  is  one  of  the  three  solid 
phases,  while  a  nonvariant  system  composed  of  three  liquid  phases 
and  vapor  is  probably  impossible. 


142  The  Phase  Rule 

Since  metals  are  not  soluble  in  ordinary  solvents,  they  are  often 
looked  upon  as  forming  a  class  by  themselves,  and  it  is  tacitly  as- 
sumed in  many  cases  that  the  behavior  of  alloys  is  not  described  by 
the  theorems  applicable  to  ordinary  chemical  phenomena.  This  is  a 
mistake.  All  the  conclusions  in  regard  to  binary  systems  which 
have  been  reached  in  the  previous  discussion  might  have  been  illus- 
trated by  taking  suitable  pairs  of  metals.  Addition  of  one  metal  to 
another  lowers  the  freezing  point  of  the  second  if  the  pure  solvent 
separates.1  If  two  metals  form  neither  compounds  nor  solid  solu- 
tions and  are  consolute  in  the  liquid  form,  the  non variant  system, 
two  solids,  solution  and  vapor,  will  be  possible  at  one  temperature 
and  pressure  only,  and  that  temperature  will  be  lower  than  the  freez- 
ing point  of  either  of  the  pure  components.  If  a  mixture  of  two 
such  metals  be  heated  until  completely  liquefied  and  the  molten  mass 
allowed  to  cool  slowly,  a  sudden  change  in  the  rate  of  cooling  will  be 
noted  at  a  temperature  which  varies  with  the  original  composition  of 
the  solution  ;  at  a  lower  temperature  the  temperature  will  remain 
constant  until  all  the  metal  has  solidified.  The  change  of  rate  oc- 
curs when  one  of  the  metals  crystallizes  from  the  solution  and  is 
therefore  a  function  of  the  concentration.  The  constant  temperature 
comes  at  the  eutectic  temperature.2  Examples  of  this  are  lead  and 
silver,  lead  and  bismuth,  tin  and  bismuth,  zinc  and  tin.  The  fact 
that  one  of  the  metals  separates  in  the  pure  state  until  a  certain  con- 
centration is  reached,  is  made  use  of  technically  in  the  Pattinson  pro- 
cess to  enrich  silver  ore.3  An  ore  rich  in  lead  and  poor  in  silver  is 
melted  and  allowed  to  cool  to  the  eutectic  temperature  when  the 
liquid  is  poured  off.  During  the  cooling,  pure  lead  crystallizes  and 
the  concentration  of  silver  increases  to  the  value  corresponding  to 
the  eutectic  alloy.  Lead  and  tin  also  form  an  eutectic  alloy  ;  but 
Kopp4  states  that  at  high  temperatures  these  metals  cease  to  be  con- 
solute  and  form  two  liquid  layers.  Unless  there  is  an  error  in  the 
determination,  this  is  analogous  to  the  case  of  diethylamine  and 


:Cf.  Tatnmann,  Zeit.  phys.  Chem.  3,  441  (1889);   Heycock  and  Neville, 
Jour.  Chem.  Soc.  55,  665  (1889)  ;  57,  376  (1890)  ;  6r,  888  (1892). 
2  Cf.  Ostwald,  Lehrbuch  I,  1025. 
»Cf.  Guthrie,  Phil.  Mag.  (5)  17,  466  (1884). 
4Liebig's  Annalen,  40,  184  (1841). 


T-U-O  Componfnfs  14^ 

water,  where  two  liquid  layers  are  formed  on  heating.  Instances 
corresponding  to  naphthalene  and  water,  where  there  may  be  equi- 
librium between  two  liquid  phases,  solid  and  vapor,  are  to  be  found 
with  lead  and  zinc,  zinc  and  bismuth,  bismuth  and  silver.1  Very 
recently  it  has  been  shown  that  zinc  and  bismuth  become  consolute 
at  about  825°  ;  zinc  and  lead  above  900°  *  Since  one  characteristic 
feature  of  a  nonvariant  system  is  that  the  temperature  remains  con- 
stant until  one  of  the  phases  has  disappeared,  it  is  clear  that  mix- 
tures of  two  metals  which  form  two  liquid  layers  will  show  two  points 
in  cooling  at  which  the  thermometer  reading  will  remain  constant  for 
a  time.  The  temperatures  of  these  points  are  independent  of  the 
initial  concentrations,  provided  two  liquid  layers  are  formed.  If 
either  of  the  components  happens  to  be  present  in  very  small  quanti- 
ties, the  nonvariant  system,  solid,  two  solutions  and  vapor,  may  not 
be  formed.  With  solutions  of  metals  it  is  often  impossible,  owing  to 
experimental  difficulties,  to  tell  by  inspection  whether  two  liquid 
phases  are  or  are  not  formed  ;  but  a  study  of  the  rate  of  cooling  will 
answer  this  question  at  once.* 

Metals  form  definite  compounds,  gold  and  aluminum,  silver  and 
aluminum  being  examples  of  this.  The  complete  curve  for  silver 
and  aluminum  has  recenth*  been  determined  by  Gautier.*  In  this 
case  the  compound,  Al,Ag,  is  stable  at  its  melting  point.  This  is 
also  true  of  the  compound  formed  of  gold  and  aluminum,  which  has 
a  melting  point  higher  than  that  of  either  component.  There  are 
probably  many  instances  of  definite  compounds  which  cease  to  be 
stable  before  the  melting  point  is  reached  ;  but  the  work  on  alloys 
has  been  done,  for  the  most  part,  in  such  an  unsatisf acton*  wa\-  that 
it  is  very  difficult  to  tell  what  the  facts  are.  To  take  a  single  illus- 
tration, the  existence  of  a  compound  XaK  is  assumed,  because  a  so- 
lution containing  these  two  metals  in  the  proportions  corresponding 
to  the  formula  solidifies  without  change  of  temperature.1  If  the 
temperature  at  which  this  takes  place  is  really  the  melting  point  of 


1  Lehmann,  Moleknlarphvsik  I,  572. 
•  Spring,  Zeit  anorg.  Chem.  13,  29  (1896). 
'Schultz,  Pogg.  Ann.  XJJ,  247  (1896). 
4  Comptes  rendus,  123,  109  ( 1896). 
5Hagen,  Wied.  Ann.  19,  436(1883). 


144  The  Phase  Rule 

this  compound,  there  must  be  two  other  solutions  which  will  also  so- 
lidify without  change  of  temperature,  yet  this  point  has  not  been 
investigated.  It  is  not  at  all  impossible  that  this  compound  does  not 
exist,  and  that  people  have  been  misled  by  the  composition  of  the 
eutectic  alloy  happening  to  coincide  very  closely  with  the  composi- 
tion of  a  definite  compound. 

Metals  also  form  solid  solutions,  and  Tammann's  experiments 
with  mercury  and  potassium,  mercury  and  sodium  point  to  the  exist- 
ence of  a  nonvariant  system  with  mercury  and  amalgam  as  the  two 
solid  phases  in  both  cases.1  Heycock  and  Neville2  observed  the  phe- 
nomenon of  a  rise  of  freezing  point  with  silver  and  cadmium,  anti- 
mony and  bismuth,  antimony  and  tin,  while  Tammann  noticed  the 
same  behavior  for  gold,  tin  and  cadmium  in  mercury,  and  Gautier3 
seems  to  have  found  a  case  with  antimony  and  aluminum,  where  the 
addition  of  the  more  fusible  to  the  less  fusible  metal  raises  the  freez- 
ing point.  The  pair  of  metals,  antimony  arid  tin,  has  been  studied 
in  detail  by  van  Bijlert4  and  by  Kiister.5  It  was  found  that  the  two 
metals  are  miscible  in  all  proportions  in  the  solid  phase,  and  that  a 
nonvariant  system  is  impossible.  Kiister  complicates  matters  un- 
necessarily in  this  paper  and  elsewhere  by  making  a  distinction  be- 
tween isomorphous  mixtures  and  solid  solutions.  With  copper  and 
nickel  the  general  form  of  the  freezing  point  curve  makes  it  probable 
that  the  solid  phases  at  one  inversion  point  are  two  sets  of  solid  solu- 
tions.6 

Some  other  determinations  by  Gautier  are  more  difficult  to  inter- 
pret because  one  does  not  know  the  probable  error  of  his  measure- 
ments. With  nickel  and  tin ,  the  curve  passes  through  a  maximum  and 
the  composition  of  the  solution  at  this  point  does  not  correspond  very 
closely  to  that  of  any  definite  compound.  Gautier  himself  thinks 
that  the  discrepancy  is  due  to  experimental  error  ;  but  in  view  of  the 
fact  that  in  the  silver  and  aluminum  series,  the  maximum  comes 


phys.  Chem.  3,  441  (1889). 
2 Jour.  Chem.  Soc.  6l,  911   (1892). 
3Comptes  rendus,  123,  109  (1896). 
4Zeit.  phys.  Chem.  8,  343  (1891). 

5  Ibid.  X2,  508  (1893). 

6  Gautier,  Comptes  rendus,  123,  172  (1896). 


Two  Components  145 

very  sharply  at  the  right  place,  it  is  more  probable  that  with  nickel 
and  tin  there  is  formed  a  solid  solution  which  can  coexist  at  one  point 
with  a  liquid  solution  of  the  same  composition.  When  nickel  is 
added  to  tin,  there  is  at  first  a  depression  of  the  freezing  point ;  but 
this  extends  over  a  very  narrow  range  of  concentrations.  Whether 
the  solid  phase  is  pure  tin  along  this  portion  of  the  curve  is  open  to 
doubt.  The  most  natural  assumption  would  be  that  the  order  of 
crystallization  was  tin,  solid  solution,  and  lastly  nickel ;  but  there  is 
no  parallel  to  this  in  any  carefully  studied  case,  so  that  it  is  not  im- 
possible that  solid  solution  separates  from  the  beginning,  and  that 
the  first  minimum  freezing  point  does  not  denote  the  existence  of  a 
nonvariant  system.  With  antimony  and  aluminum  there  are  two 
maxima,  each  higher  than  the  melting  point  of  either  component. 
It  is  probable  from  the  experiments  that  neither  is  the  melting  point 
of  a  compound,  though  it  is  unsafe  to  draw  any  conclusions  because 
it  is  stated  that  some  of  the  crystals,  after  standing,  do  not  melt 
at  the  highest  temperature  reached  by  am-  portion  of  the  freezing 
point  curve.  All  the  measurements  with  antimony  as  one  compo- 
nent need  revision,  because  Gautier  seems  to  have  worked  entirely 
with  what  is  usually  called  amorphous  antimony.1 


1  Cf.  Meslans,  Etats  allotropiques  des  corps  simples. 


10 


THREE  COMPONENTS 
CHAPTER  XI 

GENERAL  THEORY 

With  three  components,  five  phases  are  necessary  to  constitute 
a  nonvariant  system  ;  four  for  a  monovariant,  and  three  for  a  diva- 
riant  system.  It  will  simplify  matters  to  consider  first,  the  cases 
where  there  can  be  only  one  liquid  phase  and  the  solid  phases  vary 
in  composition  discontinuously,  taking  up  next  the  instances  where 
a  solid  solution  can  be  formed,  and  finishing  with  systems  in  which 
two  liquid  phases  can  be  in  equilibrium.  The  change  of  the  pres- 
sure with  the  temperature  can  be  represented  equally  well  whether 
the  system  under  consideration  be  composed  of  two,  three,  or  any 
number  of  components ;  but  a  concentration -temperature  diagram 
presents  great  difficulties  when  the  number  of  components  equals 
three.  The  problem  has  been  solved  in  quite  a  number  of  ways. 
Most  of  the  methods  give  a  solid  figure,  the  temperature  being  taken 
as  the  vertical  axis  ;  but  it  is  possible  to  tell  a  great  deal  from  the 
projections  of  the  curves  for  the  monovariant  systems  upon  a  plane, 
even  though  the  temperature  can  no  longer  be  read  directly. 

Schreinemakers1  takes  for  the  X  and  Y  axes  the  amounts  of  two 
of  the  components  in  a  constant  quantity  of  the  third.  This  is  open 
to  the  objection  that  there  is  no  place  in  the  diagram  for  an  anhy- 
drous double  salt,  nor  for  solutions  containing  very  little  of  the  third 
component.  Meyerhoffer*  has  invented  a  diagram  w7hich  has  the 
merit  of  allowing  one  to  take  the  temperature  as  one  of  the  co-ordi- 
nates. In  a  system  composed  of  two  salts  and  water,  he  measures 
the  ratio  of  one  salt  to  the  other  along  one  axis  and  the  temperature 
along  the  other.  This  is  serviceable  in  certain  cases ;  but  is  very 
limited  in  application,  since  it  neglects  the  relative  quantities  of  both 


Zeit.  phys.  Chem.  9,  67  (1892). 
Ibid.  5,  97  (1890). 


Three  Components  147 

salts  in  respect  to  the  third  component.  The  method  proposed  by 
van  Riju  van  Alkemade1  seems  to  have  no  advantage  over  the  dia- 
gram of  Schreinemakers.  Gibbs1  has  suggested  the  use  of  a  trian- 
gular diagram,  the  sum  of  the  components  being  kept  constant.  If 
we  take  an  equilateral  triangle  of  unit  height,  the  corners  of  the  tri- 
angle will  represent  the  pure  components,  and  any  point  within  the 
triangle  will  represent  some  definite  mixture  of  the  three  substances. 
The  amount  of  each  component  is  given  by  the  length  of  the  perpen- 
dicular from  the  point  to  the  side  opposite  the  vertex  corresponding 
to  that  component.  This  diagram  has  been  used  by  Thurston1  in 
some  works  on  alloys,  and  was  also  suggested  independently  by 
Stokes.*  Roozeboom5  has  used  a  modification  of  this  diagram.  He 
takes  the  isosceles  right-angle  triangle,  the  equal  sides  being  of  unit 
length.  The  advantage  of  this  arrangement  is  that  one  can  use  the 
ordinary  co-ordinate  paper  ;  but  it  is  open  to  the  objection  that  there 
is  a  different  scale  along  the  hypotenuse  from  that  along  the  sides, 
so  that  one  of  the  components  seems  to  occupy  an  exceptional  posi- 
tion. While  this  is  not  serious  in  the  case  of  two  salts  and  water 
where  the  water  is  solvent  and  the  salts  solutes,  it  is  a  disadvantage 
in  the  ternary  systems  in  which  no  such  distinction  exists  and  be- 
comes impossible  when  the  system  of  three  components  is  considered 
as  a  subdivision  of  one  containing  four.6  Roozeboom7  has  proposed 
another  form  of  triangular  diagram  which  is  distinctly  superior  to 
any  of  those  already  considered.  It  consists  of  an  equilateral  trian- 
gle with  lines  ruled  parallel  to  each  side  instead  of  perpendicular. 
The  length  of  one  side  is  taken  equal  to  unity  or  one  hundred,  and 
the  same  scale  is  used  for  the  binary  systems  in  the  side  of  the  tri- 
angle as  for  the  ternary  systems  in  the  interior.  Since  co-ordinate 
paper  can  now  be  obtained,  ruled  in  three  directions,8  this  method  of 


1Zeit.  phys.  Chem.  II,  306  (1893). 

1  Trans.  Conn.  Acad.  3,  176  (1876). 

sProc.  Am.  Ass.  «6,  114  (1877). 

4  Proc.  Roy.  Soc.  49,  174  (1891). 

5Zeit.  phys.  Chem.  12,  369  (1893). 

6  Ibid.  15,  147  (1894). 

'  Ibid.  15,  145(1894). 

8Cf.  Bancroft,  Jour.  Phys.  Chem.  I,  No.  7  (1897). 


I48 


The  Phkse  Rule 


plotting  results  will  be  used  except  in  special  cases  where  some  point 
is  to  be  brought  out  not  involving  the  variation  of  all  three  compo- 
nents. 

Since  no  single  ternary  system  has  been  studied  in  detail,  it  is 
impossible  to  select  typical  cases  illustrating  different  points,  as  was 
done  when  there  were  only  two  components.  The  special  cases, 
which  have  been  worked  out  experim entail)',  have  been  selected 
chiefly  for  their  complexity,  and  are  not  well  adapted  to  bringing  out 
the  more  general  relations.  All  that  can  be  done  is  to  point  out  the 
general  form  of  the  boundary  curves  in  the  more  simple  cases,  and 
to  go  over  what  experimental  data  there  are.  Excluding  solid  solu- 
tions we  can  have  three  different  classes  of  solid  phases,  a  pure  com- 
ponent, a  binary  compound  and  a  ternary  compound.  In  Fig.  36  are 
the  boundary  curves  for  various  possible  combinations,  starting  with 
the  simplest  case.  All  solutions  contain  one  hundred  reacting 
weights  of  A  -}-  B  -f  C.  When  the  three  components  form  no  com- 


FiG.  36. 

pounds,  the  boundary  curves  consist  of  three  lines  meeting  in  a  point. 
The  usual  form  is  represented  by  la.  The  component  A  exists  as 
solid  phase  in  the  field  marked  A,  the  components  B  and  C  in  the 
fields  marked  with  those  letters.  The  line  separating  the  fields  A 


Three  Components  149 

and  C  gives  the  compositions  of  the  solutions  which  can  be  in  equi- 
librium with  A  and  C  as  solid  phases.  Along  the  two  other  lines 
the  solid  phases  are  A  and  B,  B  and  C,  respectively.  At  the  inter- 
section of  the  three  lines  there  exists  the  non variant  system,  three 
solid  phases,  solution  and  vapor.  It  is  possible  to  say  something  in 
regard  to  the  temperature  changes.  The  corners  of  the  triangle 
represent  liquid  phases,  each  composed  of  one  of  the  three  pure  com- 
ponents in  equilibrium  with  that  component  as  solid  phase.  The 
temperature  of  the  lower  left  hand  corner  of  the  triangle  is  that  of 
the  melting  point  of  A ;  at  the  upper  corner  is  the  melting  point  of 

B.  and  at  the  lower  right  hand  corner  the  melting  point  of  C.     The 
point  on  the  side  AC  from  which  the  line  starts  toward  the  center  of 
the  triangle  represents  the  composition  of  the  eutectic  alloy  of  A  and 

C,  and  the  temperature  at  this  point  is  that  at  which  this  alloy  freezes. 
As  this  temperature  is  always  lower  in  such  cases  than  the  freezing 
point  of  either  of  the  pure  components,  the  temperature  must  fall  as 
we  pass  along  the  side  of  the  triangle  from  this  point  to  the  apex  A 
or  the  apex  C.     This  is  shown  in  the  diagram  by  the  arrow  heads 
which  point  in  the  direction  of  rising  temperature.     The  same  rea- 
soning applies  to  the  other  sides  of  the  triangle  and  the  results  are 
expressed  in  the  same  way.1 

A  theorem  by  van  Rijn  van  Alkemade*  will  serve  as  a  very 
effective  guide  in  regard  to  temperature  changes  in  the  interior  of  the 
triangle.  If  the  two  points  in  the  triangle  which  correspond  to  the 
compositions  of  two  solid  phases  be  connected  by  a  line,  the  temper- 
ature, at  which  these  same  two  phases  can  be  in  equilibrium  with  so- 
lution and  vapor,  rises  as  the  boundary  curve  approaches  this  line, 
becoming  a  maximum  at  the  intersection  though  the  boundary  curve 
often  ceases  to  be  stable  before  this  point  is  reached.  When  the  two 
solid  phases  are  two  of  the  components,  the  line  connecting  the  melt- 
ing points  is  one  of  the  sides  of  the  triangle.  It  is  therefore  dear 
that  the  temperature  must  always  rise  in  passing  along  a  boundary 
curve  to  the  side  of  the  triangle,  if  the  theorem  of  van  Alkemade  be 
right.  So  far  only  one  exception  is  known,  and  the  measurements 


Cf.  RooMboom,  Z«L  phys.  Chein.  M.  371 
'•  ibid.  n.  289  (1893). 


150  The  Phase  Rule 

in  regard  to  this  point  were  not  all  made  by  the  same  man.  Until 
the  number  of  contradictions  is  somewhat  increased  or  until  it  has 
been  shown  under  what  circumstances  the  theorem  does  not  apply,  it 
may  be  accepted  provisionally  as  accurate.  It  should  be  mentioned 
that  Schreinemakers1  has  reached  the  same  conclusion  in  regard  to 
temperature  changes  along  boundary  curves  terminating  at  the  sides 
of  the  triangle. 

In  the  particular  diagram  under  discussion,  la,  the  point  repre- 
senting the  composition  of  the  solution  in  equilibrium  with  three 
solid  phases  and  vapor  lies  within  the  triangle  formed  by  the  dotted 
lines  connecting  the  three  binary  eutectic  alloys.  While  it  is  quite 
possible  that  this  is  necessarily  the  case,  there  is  no  conclusive  proof 
of  it  and  Ib  may  represent  an  actual  system.  Here  the  point  O  lies 
outside  of  the  triangle  formed  by  the  dotted  lines  and,  in  this  par- 
ticular case,  the  nonvariant  system  formed  from  A,  B  and  C  would 
probably  exist  at  a  higher  temperature  than  either  of  the  nonvariant 
systems  formed  from  A  and  B  alone  or  B  and  C  alone.  The  direc- 
tions of  the  temperature  changes  are  shown,  as  in  the  preceding  dia- 
gram, by  the  arrow  heads.  Since  no  system  behaving  like  this  has 
yet  been  found  and  since  it  has  as  yet  no  theoretical  justification,  it 
will  be  unnecessary  to  consider  the  diagram  derived  from  it  when  the 
three  components  can  form  binary  or  ternary  compounds.  In  the 
following  discussion  only  those  cases  will  be  considered  in  which  all 
the  boundary  curves  lie  within  the  figure  formed  by  connecting  the 
points  at  which  these  curves  meet  the  sides  of  the  triangle. 

Starting  from  la  and  making  the  assumption  that  one  compound 
is  possible  we  shall  get  the  diagram  for  the  new  system  by  drawing 
a  line  from  any  point  on  any  side  to  any  point  on  an}'  boundary 
curve  or  by  drawing  a  line  from  one  boundary  curve  to  another.  To 
take  a  concrete  case,  let  us  suppose  that  A  and  C  form  a  compound 
having  the  formula  AXCV  which  we  will  represent  in  the  diagram  by 
AC.  There  are  then  three  possible  cases,  shown  in  Ila,  lib  and 
lie.  The  binary  -compound  AC  and  the  component  B  can  occur 
simultaneously  as  solid  phases  (see  Ha)  ;  AC  and  B  cannot  occur 
simultaneously  as  solid  phases  (see  lib)  ;  AC  and  B  can  occur 


1  Zeit.  phys.  Chem.  12,  73  (1893). 


simultaneously  as  solid  phases  but  AC  cannot  exist  as  a  solid  phase 
in  the  binary  system  composed  of  A  and  C  (see  He).  The  first  two 
cases  are  the  more  common  since  He  can  only  occur  when  the  com- 
ponents A  and  C  can  form  the  nonvariant  system,  two  pore  compo- 
nents, a  compound  and  vapor.  As  only  one  instance  of  this  is 
known,  in  the  double  salt  Agfflgl,,,  the  diagram  He  can  only  be 
realized  at  present  by  a  study  of  the  system,  silver  iodide,  mercuric 
iodide  and  a  thud  substance  which  melts  below  50®  and  in  which  the 
other  two  are  soluble. 

In  Ha  and  nb  the  change  of  temperature  along  the  sade  AC 
will  be  different  from  that  in  la  if  the  composition  of  the  binary 
compound  AC  is  represented  by  a  point  lying  between  the  two  inter- 
sections of  the  boundary  curves  with  that  side  of  the  triangle.  This 
has  been  assumed  to  be  the  case  in  Ila,  the  composition  of  the  double 
salt  being  given  by  the  point  H.  As  the  compound  can  then  exist 
in  eqnilibrinm  with  a  solution  of  the  same  composition  as  itself,  the 
temperature  at  His  that  of  the  melting  point  of  the  salt  and  is  higher 
than  that  at  the  intersection  of  either  boundary  curve  with  the  side 
of  the  triangle.  In  lib  the  diagram  has  been  so  drawn  that  the 
binary  compound  has  no  true  melting  point,  since  H  ties  outside  the 
field  for  AC.  In  this  case  there  win  be  only  one  temperature  mini- 
.mum  along  the  side  of  the  triangle,  at  the  point  where  the  boundary 
curve  for  A  and  AC  meets  the  side  of  the  triangle.  In  Ila  the 
boundary  curve  separating  the  field  for  B  from  that  for  AC  will  pas* 
through  a  maximum  temperature  because  it  is  cut  by  the  line  BH 
connecting  the  melting  points  of  B  end  AC.  In  Ob  the  tempera- 
ture of  the  non  variant  system  with  the  double  salt  as  one  component 
will  be  higher  than  that  of  the  nonvariant  system  with  the  compo- 
nent B  as  solid  phase  because  A  and  C  are  solid  phases  along  the 
boundary  curve  connecting  these  two  points  and  the  temperature 
therefore  rises  as  one  approaches  the  side  AC  of  the  triangle. 

In  the  diagrams  Ha  and  nb  there  is  no  difficulty  in  telling  what 

no  way  of  knowing  whether  the  compound  is  made  up  of  A  and  B, 
B  and  C,  or  A  and  C.  It  is  probable  that  in  any  actual  case  the  dia- 
gram would  not  be  symmetrical,  and  one  might  be  able  to  draw  some 


1 52  The  Phase  Rule 

conclusions  from  the  irregularity  ;  but  this  is  purely  hypothetical. 
Roozeboom1  has  pointed  out  that  the  theorem  of  van  Alkemade  re- 
quires that  the  temperature  shall  rise  along  two  sides  of  the  triangu- 
lar field  for  the  compound  to  the  common  vertex,  and  that  this  ver- 
tex points  towards  the  side  occupied  by  the  two  components  which 
form  the  compound.  In  this  way  it  is  possible  to  determine  the  con- 
stituents of  the  compound  without  an  analysis  of  the  solid  phase. 
The  results  given  by  inspection  will  be  qualitative  only.  If  the 
line  connecting  B  and  H  in  lie  cuts  the  boundary  curve  along  which 
B  and  AC  are  solid  phases,  that  point  will  be  a  maximum  tempera- 
ture for  the  boundary  curve.  If  this  is  not  the  case,  the  temperature 
will  be  highest  at  the  end  of  the  boundary  curve  near  BH.  It  is  to 
be  noticed  that  in  Ha  and  lib  there  is  one  curve  which  does  not 
reach  the  side  of  the  triangle,  and  that  in  He  there  are  three  such. 
Roozeboom  has  proposed  calling  these  ' '  middle ' '  curves  and  the 
others  "side"  curves.2 

Next  in  order  of  simplicity  is  the  ternary  system  in  which  A  and 
C  form  two  compounds,  AC  and  AC,  while  there  are  no  compounds 
containing  B.  Since  the  system  illustrated  by  lie  is  a  very  unusual 
one,  it  is  hardly  worth  while  to  consider  the  probability  of  A  and  C 
forming  another  compound.  Excluding  this,  the  diagram  for  all 
possible  systems  satisfying  the  requirements  will  be  obtained  by 
drawing  a  line  from  any  point  on  AC  to  any  one  of  the  boundary 
curves  in  Ila  or  lib.  The  three  types  of  curves  thus  obtained  are 
shown  in  Ilia,  Illb  and  IIIc.  It  should  be  mentioned  that  there  is 
always  a  change  of  direction  when  one  line  meets  another.  This  is 
not  shown  in  all  the  figures  because  the  direction  of  the  change  is  not 
always  known.  In  Ilia  either  compound  can  exist  simultaneously 
with  B  as  solid  phase  ;  in  Illb  only  one  can,  and  in  IIIc  neither  can. 
The  directions  of  the  temperature  changes  along  the  boundary  curves 
separating  the  field  for  B  from  the  fields  for  the  compounds  can  only 
be  told  when  the  compositions  of  the  compounds  are  known.  As  it 
is  improbable  that  three  components  can  form  a  ternary  compound 
when  no  two  can  form  a  binary  compound,  we  can  pass  at  once  to 


1  Zeit.  phys.  Chem.  X2,  384  (1893). 
?Ibid.  12,363(1893). 


Three  Components  153 

the  case  where  there  is  formed  one  binary  compound  AC  and  one 
ternary  compound  ABC,  illustrated  in  IV.  Since  all  substances  are 
somewhat  soluble  in  all  liquids  it  follows  that  a  ternary  com- 
pound ABC  can  exist  only  in  equilibrium  with  solutions  containing 
some  of  all  three  components,  and  that  its  field  will  be  bounded  en- 
tirely by  middle  curves.1  If  the  compound  ABC  has  a  true  melting 
point  each  curve  bounding  the  field  for  ABC  may  have  a  maximum 
temperature,  and  at  least  one  must  have.  If  it  has  not  a  true  melt- 
ing point,  the  four  middle  curves  can  not  all  have  maxima,  and  it 
may  happen  that  none  have.  There  are  three  types  of  curves  when 
there  are  two  binary  compounds  not  having  the  same  constituents. 
In  Va  there  is  one  nonvariant  system  possible  with  two  of  the  com- 
ponents as  solid  phases  ;  in  Yb  there  are  three  such,  and  in  Yc  there 
are  two  such  and  one  in  which  the  three  components  form  the  three 
solid  phases.  The  systems  possible  when  there  are  three  binary 
compounds  AC,  BC,  and  AB  will  be  found  in  the  same  way  by  draw- 
ing lines  from  the  side  AB  of  the  triangles  under  Y  to  the  other  lines 
in  the  triangles ;  but  the  number  of  hypothetical  cases  becomes  so 
large  that  it  does  not  seem  worth  while  to  take  them  up  individually. 
while  our  knowledge  of  the  subject  is  not  yet  sufficient  to  permit  of 
an  exhaustive  treatment  of  the  possible  forms  when  there  can  exist 
two  binary  compounds  AC  and  BC  together  with  one  or  more  ternary 
compounds.  At  present  we  have  no  clue  to  enable  us  to  distinguish 
between  cases  which  can  actually  exist  and  purely  artificial  forms 
which  are  the  geometrical  consequences  of  a  certain  distribution  of 
lines  in  a  plane.  Instead  of  further  speculation  in  this  direction,  it 
will  be  more  profitable  to  consider  the  experimental  data  which  have 
already  been  obtained. 

Guthrie*  has  studied  the  behavior  of  mixtures  of  potassium, 
sodium  and  lead  nitrates  so  that  it  is  possible  to  construct  a  diagram 
which  shall  represent  the  facts  with  some  approach  to  accuracy. 
This  is  done  in  Fig.  37.  The  quantities  are  expressed  in  reacting 
weights,  the  sum  of  the  three  components  being  always  equal  to  one 
hundred.  A  is  the  corner  for  potassium  nitrate  with  a  melting  point, 


Roozeboom.  ZeiL  phys.  Chem.  X2,  387  (1893). 
Phil.  Mag.  (5)  17,  472,  (1884). 


154  The  Phase  Rule 

according  to  Gtithrie,  of  320°  though  the  more  accurate  measure- 
ments of  Carnelley  and  others1  point  to  about  340°  as  the  true 
temperature  at  which  potassium  nitrate  fuses.  B  is  the  corner  for 
lead  nitrate  but  the  temperature  of  the  point  is  unknown  since  this 
substance  decomposes  before  melting.  C  is  the  corner  for  sodium 
nitrate  and  represents  a  temperature  of  305°  on  Outline's  thermo- 
meter. At  D  potassium  and  sodium  nitrates  are  present  in  the  pro- 
portions corresponding  to  the  eutectic  alloy  and  the  temperature  of 


AAA/ 

AAAAA 

AAAAAA 

AAA/VVAA 


FIG.  37. 

this  point  is  215°.  E  represents  the  eutectic  alloy  of  potassium  and 
lead  nitrates  with  a  temperature  of  207°,  while  F  is  the  correspond- 
ing point  for  lead  and  sodium  nitrates,  the  temperature  being  268°. 
Along  DO  potassium  and  sodium  nitrates  are  the  solid  phases  ; 
along  EO  potassium  and  lead  nitrates  and  along  FO  lead  and  sodium 
nitrates.  At  O  there  exists  the  notivariant  system,  potassium, 
sodium  and  lead  nitrates,  solution  and  vapor.  The  temperature  at 
which  this  system  alone  can  exist  is  186°.  In  the  field  ADOE  there 
exists  the  diva-riant  system,  solid  potassium  nitrate,  solution  and 
vapor.  In  the  solution  potassium  nitrate  is  solvent  and  the  other 
two  components  are  solutes.  In  the  field  CDOF  there  exists  the  cli- 
variant  system,  solid  sodium  nitrate,  solution  and  vapor, — sodium 

1  Landolt  and  Bernstein's  Tabellen,   148. 


Three  Components  155 

nitrate  being  solvent  in  the  solution.  In  the  field  BEOF  there 
exists,  theoretically,  the  di variant  system,  solid  lead  nitrate,  solution 
and  vapor.  Practically  this  is  interfered  with  by  the  partial  decom- 
position of  the  lead  nitrate.  This  is  a  secondary  phenomenon  and 
makes  the  selection  of  lead  nitrate  as  one  of  the  three  components  an 
unfortunate  one. 

It  will  be  interesting  to  consider  the  general  form  of  the  isothermal 
curves  for  a  system  like  this  one,  made  up  of  three  components 
which  form  no  compound.  It  is  explicitly  assumed,  for  purpose  of 
discussion,  that  lead  nitrate  is  stable.  The  dotted  lines  X  X  repre- 
sent the  general  form  of  the  isotherm  for  a  temperature  higher  than 
that  of  any  of  the  binary  eutectics  and  lower  than  the  melting  point 
of  the  most  fusible  component,  in  other  words  for  a  temperature  of 
about  290°.  The  lines  are  not  really  straight  lines  an\-  more  than 
DO,  EO  and  FO  are ;  but  they  are  drawn  approximately  straight 
because  the  only  determination  we  have  in  the  interior  of  the  tri- 
angle is  that  of  the  point  O.  It  will  be  noticed  that  this  particular 
isotherm  consists  of  three  parts1  because  the  temperature  rises  in 
both  directions  along  the  sides  of  the  triangle  from  the  binary 
eutectics.  At  the  temperature  of  the  eutectic  alloy  of  lead  and 
sodium  nitrates.  268°,  the  two  branches  of  the  curve  come  together 
on  the  BC  side  of  the  triangle  and  the  isotherm  for  268°  has  the 
general  form  shown  by  the  dotted  lines  YY  and  YFY.  At  a  lower 
temperature  two  branches  of  the  isotherm  would  meet  at  some  point 
on  the  curve  OF  without  ever  reaching  the  side  BC.  At  a  tempera- 
ture lower  than  the  freezing  points  of  any  of  the  binary  eutectics.  at 
195°  for  instance,  the  general  form  of  the  isotherm  will  be  given  by 
the  dotted  lines  ZZZ.  The  isotherm  has  become  a  clos:d  curve  which 
will  diminish  in  area  as  the  temperature  falls  until,  at  186°,  it  will 
consist  only  of  the  point  O.  At  still  lower  temperatures  there  will 
be  no  isotherm  because  there  will  be  no  solution  and  the  solid  phases 
do  not  react  each  with  the  other.  The  isotherms  approach  the  point 
O  with  falling  temperature  and  recede  from  it  with  rising  tempera- 
ture. When  the  temperature  rises  above  the  melting  point  of  the 


1  Roozeboom  treats  each  part  as  a  whole,  speaking  of  the  isotherm  for  a 
given  solid  phase  ;  bat,  this  is  not  to  be  commended. 


156  The  Phase  Rule 

most  fusible  compound,  the  isotherm  consists  of  two  branches  only 
because  a  component  can  not  exist  as  solid  phase  at  a  temperature 
above  its  own  melting  point.  At  a  temperature  below  the  melting 
point  of  the  least  fusible  component  and  above  those  of  the  other 
two,  the  isotherm  has  only  one  branch  while  it  is  reduced  to  a  point 
at  the  melting  point  of  the  least  fusible  component.  Above  this 
last  temperature  there  can  be  no  isotherm  because  the  three  com- 
ponents are  liquids  and  all  consolute,  by  hypothesis,  so  that  a  di- 
variaut  system  is  impossible. 

In  considering  the  change  of  the  isotherms  with  the  temperature 
for  a  ternary  system  wyhich  permits  of  no  compounds  and  no  second 
solution  phase,  it  is  necessary  to  distinguish  two  cases.  The  melting 
points  of  the  pure  components  may  each  be  higher  than  the  melting 
point  of  any  of  the  binary  eutectics.  This  is  true  with  potassium, 
sodium  and  lead  nitrates,  and  is  the  rule  when  the  three  components 
are  similar  in  nature.  One  of  the  components  may  melt  at  a  lower 
temperature  than  the  binary  eutectic  formed  from  the  other  two  com- 
ponents. This  would  occur  with  ice  and  two  anhydrous  salts,  and 
is  the  rule  whenever  the  freezing  point  of  one  component  lies  far  be- 
low the  freezing  point  of  either  of  the  other  two.  In  the  first  case, 
the  isotherm  for  a  monovariant  system  will  pass  through  the  follow- 
ing forms  as  the  temperature  falls :  a  point,  a  branch  and  a  point, 
two  branches,  two  branches  and  a  point,  three  branches,  an  isolated 
and  two  connecting  branches,  three  connecting  branches,  a  closed 
curve,  a  point.  In  the  second  case,  the  order  wilt  be :  a  point,  a 
branch,  a  branch  and  a  point,  two  branches,  two  connecting 
branches,  two  connecting  branches  and  a  point,  two  connecting 
branches  and  a  branch,  three  connecting  branches,  a  closed  curve,  a 
point.  The  limiting  points  for  the  isotherms  are  the  highest  tem- 
perature at  which  any  solid  phase  can  exist  in  equilibrium  with  solu- 
tion and  vapor  and  the  lowest  temperature  at  which  this  is  possible. 

In  the  case  of  the  melted  nitrates,  the  field  in  which  any  one  of 
the  components  is  solvent  is  identical  with  the  field  in  which  that 
component  can  occur  as  solid  phase  ;  but  this  is  not  necessarily  true. 
As  a  rule  this  will  be  so  only  when  the  melting  points  are  not  too 
far  apart.  It  would  be  an  interesting  point  to  consider  whether  the 
change  came  when  the  melting  point  of  one  of  the  components  fell 


Three  Gmmpememts 


below  that  of  the  entectic  ft 


the  other  tiro;  bat  there 
The  most  striking 


as  jet  no  data  on  which  to  base : 
of  a  substance  existing  as  sofid  phase  without  being  the  solvent  in 
the  solution  is  to  be  found  in  the  equilibrium  between  two  salts  and 
water.  The  two  salts  must  have  the  same  acid  or  the  same  basic 
radicle,  otherwise  the  number  of  components  is  no  longer  three.  In 
Fig.  38  is  the  graphical  representation  off  what  few  data  are  accessible 
for  the  system  potassium  chloride,  potassium  nitrate  and  water. 


Only  a  small  portion  of  the  diagram  isshown,  as  the  whole  would 
require  a  triangle  five  times  as  long  on  the  side.  The  concentrations 
are  expressed  in  reacting  weights,  the  sum  of  the  three  components 
being  always  equal  to  one  hundred  reacting  weights.  At  D  is  the 
cryohydoc  point  for  ke  and  potassium  chloride1,  the 


Ibg.  is}  x*  149,  («***- 


158        ,  The  Phase  Rule 

being—  10.7°.  At  E,  —3°,  is  the  cryohydric  point  for  ice  and 
potassium  nitrate.  Along  EO  ice  and  potassium  nitrate  are  solid 
phases  ;  along  DO  ice  and  potassium  chloride  and  along  FO  potas- 
sium nitrate  and  sodium  nitrate.  The  concentration  and  tempera- 
ture corresponding  to  the  point  F  have  not  been  determined  experi- 
mentally. Fvtard1  has  stated  that  the  line  OF  terminates  at  the  melt- 
ing point  of  the  more  fusible  salt.  In  other  words,  in  this  particular 
case,  the  amount  of  water  in  the  solution  in  equilibrium  with  solid 
potassium  chloride  and  nitrate  will  become  zero  at  the  melting  point 
of  potassium  nitrate.  This  is  entirely  wrong.  The  .curve  OF  ter- 
minates at  the  temperature  of  the  eutectic  alloy  formed  from  the  two 
salts,  a  temperature  which  is  necessarily  lower  than  the  melting 
point  of  potassium  nitrate.  Curiously  enough, '^tard  has  an  inkling 
of  the  truth  in  one  case2,  -but  it  is  not  sufficient  to  make  him  modify 
his  erroneous  hypothesis.  At  O  there  exists  the  non variant  system, 
ice,  potassium  chloride,  potassium  nitrate,  solution  and  vapor,  five 
phases.  The  temperature  is  —  11.4°.  In  the  field  ADOE  ice  is 
solid  phase  ;  in  BEOF  potassium  nitrate  and  in  CDOF  potassium 
chloride.  Water,  however,  is  solvent  in  the  whole  of  the  first  field 
and  in  parts  of  the  two  others.  Where  the  change  takes  place  is  not 
known  ;  but  there  would  be  a  line  starting  from  the  point  where  EB 
ceases  to  be  a  solubility  curve  and  becomes  a  fusion  curve  and  this 
line  would  cut  OF  at  some  point.  On  one  side  of  this  hypothetical 
line  water  would  be  solvent ;  on  the  other  potassium  nitrate.  Simi- 
larly a  line  could  be  drawn  from  a  point  on  DC  to  OF,  such  that  on 
one  side  of  it  water  would  be  solvent,  on  the  other  potassium 
chloride.  The  determination  of  these  lines  is  one  of  the  interesting 
problems  in-theoretical  chemistry. 

The  isotherm  for  20°  has  the  general  form  shown  by  the  dotted 
lines  XXX.3  It  consists  of  two  connected  branches.  A  third  branch 
is  impossible  because  the  temperature  is  above  the  melting  point  of 
ice.  The  isotherm  for  o°  consists  of  two  connected  branches  and  a 
point,  illustrated  by  the  dotted  lines  YYY  and  the  point  A.  At  a 


'Coniptes  rendus,  109,  740  (1889),  Ann.  chiin.  phys.  (7)  3,  275  (1894). 

9 Ibid.  (7)3,284,  (1894). 

"Nicol,  Phil.  Mag.  (5)  31,  369  (1891). 


Three  Components  159 

little  lower  temperature  the  point  will  become  a  line  across  the  cor- 
ner and  at  temperatures  below  — 10.7°  the  isotherm  will  have  the 
form  of  a  closed  curve.  In  all  cases  where  the  isotherm  crosses  a 
boundary  curve,  there  is  a  change  of  direction  because  there  is  a 
change  in  the  nature  of  the  solid  phase.  There  is  also  a  change  of 
direction  when  the  isotherm  cuts  the  line  separating  the  field  for  one 
component  as  solvent  from  the  field  in  which  another  component  is 
solvent.  It  matters  not  whether  one  is  dealing  with  a  system  of  two, 
of  three  or  of  any  number  of  components.1  The  composition  of  the 
saturated  solution  changes  suddenly  when  the  nature  of  the  solid 
phase  or  of  the  solution  changes.  The  change  of  solvent  cannot  be 
sh'own  at  ordinary  temperatures  with  the  system,  potassium  chloride, 
potassium  nitrate  and  water,  because  the  salts  do  not  act  as  solvent. 
In  the  system,  potassium  chloride,  alcohol  and  water,  we  can  pass 
from  a  solution  with  potassium  chloride  as  solid  phase  and  water  as 
solvent  to  one  with  potassium  chloride  as  solid  phase  and  alcohol  as 
solvent.  In  Fig.  39  is  given  the  isotherm  for  sodium  chloride, 
sodium  nitrate  and  water  at  20°  taken  from  XicolV  measurements 
and  the  isotherm  for  sodium  chloride,  alcohol  and  water  at  30 =  taken 
from  Bathrick's3  determinations.  The  diagram  is  the  one  recom- 
mended by  Schreinemakers,  the  co-ordinates  being  the  concentra- 
tions of  two  of  the  components  in  a  constant  quantity  of  water,  in 
this  case  grams  of  the  salts  and  of  alcohol  in  one  hundred  grams  of 
water.  The  only  change  is  that  the  logarithms  of  these  concentra- 
tions are  taken  instead  of  the  concentrations  themselves.  The 
abscissae  are  the  logarithms  of  the  concentrations  of  sodium  chloride 
and  the  ordinates  the  logarithms  of  the  concentrations  of  sodium 
nitrate  and  of  alcohol.  The  scale  for  the  alcohol  is  one-half  that  for 
the  sodium  nitrate. 

The  similarity  of  the  two  curves  is  very  striking.  Along  AB 
sodium  chloride  is  solid  phase  and  along  BC  sodium  nitrate,  water 
being  solvent  along  the  whole  length  of  the  curve.  There  is  a  dis- 
tinct break  at  B  where  the  change  occurs  in  the  solute  with  respect 


1  Bancroft,  Phys.  RCT.  3,  204  (1895) ;  Jour  Phys.  Chem.  I,  36  (1896). 
'Phil.  Mag.  (5)  JK,  369  (1891). 
'Jonr.  Phys.  Chem.  I.  No.  3,  1896. 


i6o 


The  Phase  Rule 


to  which  the  solution  is  saturated.     Along  the  whole  of  the  curve 
A'B'C',   sodium    chloride  is  solid  phase  ;  but  water  is  solvent  along 


FIG.  39. 


Three  Components 


and  alcohol  along  B^.  There  is  a  distinct  break  at  V  where 
the  nature  of  the  solvent  changes.  The  analogy  between  the  two 
cases  is  complete. 

Although  no  experimental  work  has  yet  been  done  illustrating 
the  other  cases  which  are  represented  in  Fig.  36,  it  will  be  well  to 
consider  the  general  form  of  the  isotherms  in  one  or  two  particular 
instances  as  a  guide  to  future  investigators.  For  the  sake  of  sim- 
plicity it  will  be  assumed  that  the  theorem  of  van  Alkemade  holds 
good,  that  the  temperature  rises  along  a  boundary  curve  in  the  direc- 
tion towards  the  line  connecting  the  melting  points  of  the  two  solid 
phases  which  exist  along  the  boundary  curve  in  question.  There 
are  so  few  exceptions  to  this  rule  that  it  is  not  worth  while  to  con- 
sider them  until  they  have  been  studied  more  fully,  both  experi- 
mentally and  theoretically.  In  Fig.  40  are  some  of  the  cases  from 


FIG.  40. 


1 62  The  Phase  Rule 

Fig.  36  with  a  few  characteristic  isotherms  sketched  in.  la  is  the 
diagram  for  a  system  forming  one  binary  compound  AC  which  is  sta- 
ble at  its  melting  point,  and  which  can  coexist  with  any  of  the 
three  components  as  solid  phase.  In  the  field  DOKE  the  binary 
compound  is  solid  phase.  It  is  not  known  what  is  solvent  in  this 
field  ;  but  it  is  probable  that  the  component  A  assumes  that  function 
in  one  part  of  the  field  and  the  component  C  over  the  rest  of  it. 
There  seems  no  reason  to  assume  that  B  is  solvent  anywhere  outside 
of  the  field  BGOKF.  It  should  be  clearly  understood  that  this  ap- 
plies only  to  those  cases  where  A  and  C  are  solvents  in  the  respective 
fields  in  which  they  occur  as  solid  phases.  If  A  is  solvent  in  part  of 
the  field  in  which  C  is  solid  phase,  A  is  also  solvent  in  the  whole  of 
the  field  occupied  by  AC.  This  case  will  occur  with  two  salts  and 
water,  the  only  binary  compound  being  a  hydrate  with  a  low  inver- 
sion temperature. 

At  temperatures  just  below  the  melting  point  of  the  binary  com- 
pound the  isotherm  will  usually  have  the  form  represented  by  the 
lines  marked  i,  four  isolated  branches,  one  of  them  forming  a  semi- 
circular curve  round  H,  themelting  point  of  the  compound .  This  last 
branch  may  curve  in  at  the  bottom.1  If  the  binary  compound  has  a 
higher  melting  point  than  any  of  the  components, *  the  first  isotherm 
will  consist  of  the  curve  round  H  and  the  other  three  branches  will 
appear  one  by  one  as  the  temperature  falls.  At  a  lower  temperature 
the  isotherm  marked  2  has  become  a  closed  curve  made  up  of  four 
parts.  It  is  clear  that  the  isotherm  first  becomes  a  closed  curve  at 
the  lowest  temperature  at  which  any  boundary  curve  meets  the  side 
of  the  triangle.  This  is  a  necessary  geometrical  consequence  of  this 
form  of  diagram  and  has  no  theoretical  significance.  The  isotherm 
marked  3,  is  the  first  to  meet  the  line  OK.  The  general  form  is  that 
of  a  figure  eight.  The  contact  occurs  at  the  point  where  the  line 
BH  cuts  OK  and  represents  the  maximum  temperature  at  which  AC 
and  B  can  be  in  equilibrium  together  with  the  solution  and  vapor. 


1  Cf.  Rooieboom,  Zeit.  phys.  Cbem.  X5,  602  (1894). 

JThe  compound  AuAl2  has  a  higher  mehing  point  than  pure  gold  so  that  a 
ternary  system  satisfying  these  conditions  is  possible.  Cf.  Roberts- Austen, 
Proc.  Roy.  Soc.  50,  367  (1891). 


Three  Components  163 

If  the  temperature  be  raised  at  all,  both  solid  phases  will  change  into 
solution  until  one  of  them  has  disappeared.  Which  will  disappear 
first  depends  on  the  relative  quantities  of  the  two  solid  phases.  This 
maximum  temperature  has  been  called  the  fusion  point  of  two  solid 
phases.1  It  should  be  kept  in  mind  that  this  intersection  is  a  maxi- 
mum temperature  for  the  line  OK  but  a  minimum  temperature  for 
BH.  At  yet  lower  temperatures,  the  isotherm  becomes  two  detached , 
closed  curves  contracting,  one  to  the  point  O,  the  other  to  the  point 
K.  If  the  temperature  of  K  is  higher  than  that  of  O,  the  final  re- 
sult will  be  a  single  closed  curve  contracting  to  O  as  a  vanishing 
point.  If  the  temperature  of  K  is  lower  than  that  of  O  the  last  iso- 
therm possible  will  consist  of  a  point  at  K. 

In  Ib  the  melting  point  H  of  the  binary  compound  lies  outside  the 
field  for  AC  and  does  not  represent  a  state  of  stable  equilibrium. 
Under  these  circumstances,  the  temperature  will  rise  continuously  as 
the  system  passes  from  O  to  K  since  this  is  in  the  direction  towards 
the  line  BH.  In  this  diagram  the  melting  point  of  A  is  assumed  to 
be  lower  than  the  temperature  of  the  point  K  and  the  temperature  of 
E  to  be  lower  than  that  of  F.  The  first  isotherm  is  for  a  tempera- 
ture between  those  of  the  points  E  and  F,  the  second  for  a  tempera- 
ture between  K  and  E  and  the  third  for  the  temperature  of  the  point 
K.  Here  for  the  first  time,  the  part  of  the  isotherm  along  which  B 
is  solid  phase  meets  the  part  of  the  isotherm  along  which  AC  is  solid 
phase.  The  point  of  intersection  is  no  longer  somewhere  in  the 
middle  of  the  line  OK  nor  is  it  on  the  line  connecting  B  and  H  ;  but 
it  is  the  highest  temperature  possible  for  the  stable  curve  OK  and 
the  minimum  temperature  for  the  line  BKH.  If  the  appearance  at 
K  of  C  as  solid  phase  can  be  prevented  and  the  curve  OK  followed 
further  it  will  be  found  that  the  temperature  will  continue  to  rise, 
passing  through  a  maximum  at  the  intersection  of  the  prolongation 
of  OK  with  DH.  At  lower  temperatures  the  isotherm  becomes  a 
closed  curve  contracting  to  the  point  O. 

In  II  the  fields  for  the  binary  compound  and  for  the  third  com- 
ponent are  entirely  separated.2  The  first  isotherm  is  taken  at  the 


1  Roozeboom,  Zeit.  pbys.  Chem.  15,  616  (1893). 

2  It  is  not  at  all  certain  that  such  a  case  can  actually  occur. 


1 64  The  Phase  Rule 

temperature  of  the  point  E  to  illustrate  the  fact  that  the  temperature 
differences  HE  and  HO  are  usually  very  unequal.  The  second 
isotherm  in  the  diagram  is  a  closed  curve  made  up  of  four  parts  and 
the  third  a  closed  curve  with  three  parts.  This  latter  vanishes  at  the 
point  O,  because  the  temperature  rises  along  the  line  OK  according 
to  the  theorem  of  van  Alkemade. 

In  Ilia  there  are  two  binary  compounds  represented  by  the  con- 
ventional formulas  AC  and  BC.  Both  have  stable  melting  points 
and  it  is  assumed  that  the  line  AR,  if  drawn,  would  cut  OP  at  some 
point  and  that  the  line  RH  would  cut  PK.  These  two  intersections 
will  be  maximum  temperatures  for  OP  and  PK  respectively.  It  is  also 
assumed  that  the  maximum  temperature  for  OP  is  lower  than  the 
maximum  temperature  for  PK,  lower  than  the  temperature  of  F  and 
higher  than  the  temperature  of  K.  Under  these  circumstances  the 
isotherm  for  the  maximum  temperature  for  OP  has  the  general  form 
of  a  figure  eight  round  O  and  P  and  a  detached  closed  curve  round 
K  as  represented  in  the  diagram.  In  IHb  the  conditions  are  the 
same  except  that  there  is  no  maximum  for  OP,  the  temperature 
rising  continuously  from  O  to  P.  The  isotherms  in  the  diagram  are 
for  the  maximum  temperature  for  PK  and  for  the  temperature  of 
the  point  P.  In  this  diagram  and  the  preceding  one  the  final  iso- 
therm is  the  point  O.  In  IIIc  neither  binary  compound  has  a  stable 
melting  point  and  the  melting  point  of  A  is  assumed  to  be  lower  than 
the  temperature  of  the  points  F  and  N.  The  first  isotherm,  marked 
i,  is  that  passing  through  the  point  K  and  the  second  the  correspond- 
ing one  for  the  point  P.  If  the  melting  point  R  had  been  situated 
very  close  to  the  apex  of  the  triangle  the  temperature  would  have 
risen  in  passing  from  P  to  O  instead  of  falling  as  is  now  the  case  and 
the  isotherm  at  P  would  have  been  merely  a  point. 

When  the  curves  for  the  monovariant  systems  do  not  pass 
through  a  maximum  temperature,  the  parts  of  the  isotherms  in  the 
adjoining  fields  first  come  in  contact  at  the  quintuple  point  terminat- 
ing the  curve  at  the  higher  temperature  end.  It  has  already  been 
shown  in  discussing  la  that  when  there  is  a  temperature  maximum 
the  two  parts  of  the  isotherm  become  tangent  at  that  point.  At 
lower  temperatures  they  will  meet  at  an  angle. 


CHAPTER  XII 

TWO  SALTS   AND   WATER 

In  going  over  the  systems  which  have  been  worked  out  experi- 
mentally, it  will  be  best  to  begin  with  the  one  composed  of  mag- 
nesium sulfate,  potassium  sulfate  and  water,  which  has  been  studied 
by  van  der  Heide1.  Only  a  small  portion  of  the  entire  field  has  been 
examined  and  this  is  reproduced  in  Fig.  41.  The  concentrations  are 


FIG.  41. 

expressed  in  reacting  weights,  the  sum  of  the  three  being  always 
equal  to  one  hundred.     G  is  the  cryohydric  point  for  potassium  sul- 


•Zeit  phys.  Chem.  12,  416,  (1893). 


1 66  The  Phase  Rule 

fate  and  ice,  the  temperature  being—  1.2°.  H  is  the  correspond- 
ing point  for  magnesium  sulfate  heptahydrate  and  ice,  the  tem- 
perature being  —  6°  owing  to  the  greater  solubility  of  the  magnesium 
sulfate.  Along  GA  ice  and  potassium  sulfate  are  solid  phases  and 
along  HA  ice  and  magnesium  sulfate  heptahydrate.  At  A,  —  4-5°, 
there  exists  the  non variant  system,  ice,  magnesium  sulfate  with 
seven  of  water,  potassium  sulfate,  solution  and  vapor.  The  tem- 
perature is  higher  than  that  of  the  point  H,  contrary  to  the  rule  of 
Schreinemakers1  and  van  Alkemade.  The  reason  for  this  is  that  po- 
tassium sulfate  precipitates  the  heptahydrate  of  magnesium  sulfate 
in  such  quantities  that  the  solution  actually  becomes  more  dilute  in 
passing  from  H  to  A,  there  being  three  and  seven-tenths  units  of 
magnesium  sulfate  in  solution  at  H  while  the  sum  of  the  two  sulfates 
at  A  is  only  two  and  two-tenths  units.  Of  course,  potassium  sulfate 
has  a  greater  effect  in  lowering  the  vapor  pressure  of  a  solution  than 
magnesium  sulfate  ;  but  this  is  more  than  balanced  by  the  difference 
in  concentration  so  that  the  vapor  pressure,  and  therefore  the  freez- 
ing point,  of  the  solution  at  A  is  higher  than  at  H.  This  is  an  in- 
teresting instance  of  the  inapplicability  of  one  of  Ostwald's  most  im- 
portant theorems.  Ice,  potassium  sulfate  and  the  heptahydrate  of 
magnesium  sulphate  are  in  equilibrium  at—  4.5°  and  therefore  ice 
and  the  magnesium  salt  should  be  in  stable  equilibrium.  As  a  mat- 
ter of  fact,  unless  there  is  an  error  in  the  determinations,  a  mixture 
of  these  two  substances  will  liquefy  at  this  temperature. 

Along  the  curve  B  the  solid  phases  are  the  two  sulfates.  At  B, 
—  3°,  the  two  sulfates  unite  to  form  a  ternary  compound  having  the 
formula  K2Mg(SO4)2  6H2O.  We  have  thus  at  B,  a  new  nonvariant 
system,  with  potassium  sulfate,  magnesium  sulfate  heptahydrate  and 
the  hydrated  double  salt  as  solid  phases.  Along  BC  the  solid  phases 
are  the  hydrated  double  salt  and  magnesium  sulfate  heptahydrate. 
At  C,  47.2°,  the  salt  with  seven  of  water  passes  into  that  with  six  of 
water  and  there  is  formed  another  nonvariant  system  with  the  hepta- 
hydrate, the  hexahydrate  and  hydrated  double  salt  as  solid  phases. 
Along  CK  the  solid  phases  are  the  heptahydrate  and  the  hexa- 
hydrate of  magnesium  sulfate ;  K,  48.2°,  represents  the  temperature 

1  Zeit.  phys.  Chem.  12,  77,  (1893). 


Three  Components  167 

and  concentration  at  which  magnesium  sulfate  with  seven  and  six  of 
water  are  in  equilibrium  with  solution  and  vapor  when  no  potassium 
snlfate  is  present.  In  other  words  it  is  the  inversion  point  for  the 
two  hydrates  in  the  binary  system,  magnesium  sulfate  and  water. 
Along  CD  the  solid  phases  are  the  hydrated  doable  salt  and  mag- 
nesium sulfate  with  six  of  water.1  At  D.  72°,  another  solid  phase 
appears  in  the  form  of  a  second  hydrated  double  salt  with  the 
formula  K,Mg(  SOt),  4H4O.  Along  DL  the  solid  phases  are  the 
new  hydrated  double  salt  and  the  hexahydrate  of  magnesium  sulfate. 
This  curve  has  not  been  followed  very  far  ;  but  an  experiment  in  a 
sealed  tube  showed  that  at  a  temperature  of  106°  a  change  takes 
place  presumably  of  the  hexahydrate  into  a  monohydrate.  Along 
DE  the  solid  phases  are  the  two  hydrated  double  salts.  At  E,  92°, 
the  solution  becomes  saturated  in  respect  to  potassium  sulfate  form- 
ing yet  another  nonvariant  system.  Along  EF  the  solid  phases  are 
potassium  sulfate  and  the  salt  KjMg(SOt\  4^0.  At  some  point  on 
this  curve  the  hydrated  double  salt  will  lose  water  changing  into 
something  else  ;  but  it  is  not  known  what  this  change  is  nor  at  what 
temperature  it  takes  place.  Along  EB  the  solid  phases  are  potassium 
sulfate  and  the  hydrated  double  salt  K,Mg(SO4\  6ILO.  In  the 
field  to  the  left  of  HAC  ice  is  the  solid  phase ;  in  the  field  HABCK 
magnesium  sulfate  with  seven  of  water.  The  hexahydrate  exists  in 
the  field  bounded  by  KCDL  and  the  undetermined  line  for  the  mono- 
hydrate.  Potassium  sulfate  occurs  as  solid  phase  in  the  field  bounded 
by  GABEF  and  other  lines  not  yet  determined.  The  field  for 
K.Mg-  SOt \  6H,O  is  entirely  determined,  being  contained  within 
the  closed  figure  BCDEB.  On  the  other  hand,  the  field  for  the 
other  hydrated  double  salt  K.XIgtSO,),  4HtO  is  bounded  on  the  left 
by  the  lines  LDEF  while  the  right  hand  boundary  is  unknown. 

If  we  draw  the  dotted  line  XX  through  the  lower  left  hand  cor- 
ner of  the  triangle  and  the  middle  of  the  opposite  side,  this  line 
represents  a  series  of  solutions  in  which  potassium  and  magnesium 
sulfates  are  present  in  equivalent  quantities.  A  point  P  on  this  line 


1  There  is  a  second  modification  of  magnesium  sulfate  herahydrale  ;  bat  it 
is  labile  at  all  temperatures  and  is  not  considered  here. 
ZeiL  phys.  Chem    ia,  421  (1893). 


1 68  The  Phase  Rule 

which  does  not  happen  to  fall  within  the  limits  of  the  present  dia- 
gram would  represent  a  solution  having  the  same  composition  as  the 
hydrated  double  salt  with  six  of  wyater  while  yet  another  point  Q 
farther  out  would  represent  a  solution  having  the  same  composition 
as  the  second  hydrated  double  salt,  the  one  with  four  of  water. 
Whether  this  second  point  lies  within  the  field  for  the  salt  K2MgSO4 
4.H2O  is  not  known  because  the  boundaries  of  this  field  have  not  yet 
been  fully  determined  ;  but  it  is  improbable  that  this  is  the  case. 
We  know,  however,  that  the  solution  having  a  composition  repre- 
sented by  the  formula  K2Mg(SO4)2  6H2O  does  not  lie  within  the  field 
in  which  that  compound  can  exist  as  a  solid  phase.  In  other  words 
this  particular  hydrated  double  salt  can  not  exist  in  equilibrium 
with  a  solution  having  the  same  composition  as  itself  and  therefore 
has  not  a  true  melting  point.  From  the  position  of  the  dotted  line 
XX  there  are  other  conclusions  to  be  drawn.  It  does  not  cut  the 
field  for  K2Mg(SO4)s  6H2O  at  any  point  and  therefore  this  salt  can 
not  be  in  equilibrium  with  any  solution  in  which  potassium  sulfate 
and  magnesium  sulfate  are  present  in  equivalent  quantities.  Addi- 
tion of  water  to  this  salt  will  therefore  decompose  it,  dissolving  out 
an  excess  of  one  component  and  leaving  some  of  the  other  as  solid 
phase.1  In  this  particular  case  addition  of  water  to  the  hydrated 
double  salt  at  any  temperature  between  — 3°  and  92°  will  cause  a 
partial  decomposition  with  formation  of  potassium  sulfate  as  solid 
phase,  the  solution  having  the  concentration  corresponding  to  some 
point  on  the  curve  BE.  Further  addition  of  water  will  cause  increased 
formation  of  potassium  sulfate  and  an  increased  amount  of  solution, 
the  concentration  remaining  constant  until  the  whole  of  the  hydrated 
double  salt  has  disappeared,  leaving  the  divariant  system,  potassium 
sulfate,  solution  and  vapor.  On  adding  more  water  the  potassium  sul- 
fate will  dissolve  giving  finally  a  series  of  unsaturated  solutions  in  which 
the  potassium  and  magnesium  sulfates  are  at  last  present  in  equiva- 
lent quantities.  With  the  hydrated  double  salt,  K2Mg(SO4)2  4H2O, 
the  case  may  be  different.  If  the  dotted  line  XX  cuts  the  line  EF, 
as  it  seems  to  in  the  diagram,  this  ternary  compound  will  not  be  de- 


1  Cf.  Roozeboom,  Zeit.   phys.  Chem.  2,  520  (1888)  ;  Sclweinemakers,  Ibid, 
9,  75  (1892). 


Three  Components  169 

composed  by  water.  Unfortunately  the  line  EF  has  only  been  fol- 
lowed a  very  short  way  and  it  may  come  to  an  end  before  it  reaches 
the  point  F  in  the  diagram.  All  that  can  be  said  is  that  if  the  dotted 
line  cuts  the  field  for  the  hydrated  double  salt,  KjMg(SO4),  4H2O, 
this  compound  will  not  be  decomposed  by  water  ;  otherwise  it  will. 
It  is  not  of  much  importance  either  way  as  there  are  cases  known 
where  the  ternary  compound  is  not  decomposed  by  water. 

In  regard  to  the  changes  of  direction  at  the  inversion  points, 
there  is  a  theorem  by  Meyerhoffer1  which  has  a  certain  qualitative 
value.  It  is  that  the  solubility  curve  for  the  solid  phase  which  does 
not  disappear  has  no  break  at  the  inversion  temperature.  While  it 
is  necessarily  true  that  the  concentration  curve  for  the  component 
which  disappears  as  solid  phase  will  always  have  a  break  at  the  in- 
version point,  the  converse  of  this,  which  is  the  theorem  of  Meyer- 
hoffer, will  never  be  true  exactly  though  the  approximation  may  be 
very  close.  A  change  in  the  nature  of  the  second  solid  phase  will 
necessarily  affect  the  solubility  of  the  first  because  it  is  impossible 
that  two  different  substances  can  cause  the  same  precipitation,  posi- 
tive or  negative,  of  another  body  over  a  range  of  temperatures.  On 
the  other  hand,  this  difference  of  effect  will  be  relatively  small  and 
can  usually  be  ignored.  The  great  difficulty  in  regard  to  this 
theorem  of  Meyerhoffer  is  that  the  way  in  which  it  is  deduced  would 
seem  to  imply  that  the  concentrations  should  be  given  as  amounts  of 
each  salt  in  a  constant  quantity  of  water  or  in  a  constant  quantity  of 
water  plus  that  salt  whereas  Meyerhoffer  takes  the  amounts  of  each 
salt  in  one  hundred  grams  of  the  solution  containing  water  and  both 
salts.  He  is  not  consistent  even  in  this  because  the  data  of  Rooze- 
boom2  are  advanced  as  proofs  of  the  theorem,  though  these  are  ex- 
pressed as  reacting  weights  of  each  salt  in  one  hundred  reacting 
weights  of  water. 

This  theorem  can  be  applied  in  the  system  under  discussion. 
Along  the  line  BCD  potassium  magnesium  sulfate  with  six  of  water 
is  one  of  the  solid  phases,  magnesium  sulfate  with  seven  of  water 
being  the  other  solid  phase  along  BC  and  magnesium  sulfate  with 


'Zeit.  phys.  Chem.  5,  120  (1890). 
-  Ibid.  2,  518  (  iSSS). 


170 


The  Phase  Rule 


six  of  water  along  CD.  It  is  found  that  the  amount  of  potassium 
sulfate  in  solution  changes  practically  continuously  with  the  tempera- 
ture while  the  amount  of  magnesium  sulfate  in  the  unit  quantity  of 
solution  changes  discontinuously  as  the  temperature  becomes  47.2°, 
at  the  point  C.  Along  the  curve  GDI,  the  concentration  of  potassium 
sulfate  changes  discontinuously  at  D  while  that  of  magnesium  sulfate 
does  not.  Along  BEF  the  reverse  is  the  case.  On  the  other  nand, 
Meyerhoffer  seems  to  have  found,  in  the  system,  sodium  and  mag- 
nesium sulfates,  and  water,  a  case  where  both  curves  show  a  break. 
This  however  is  not  well  established.1 

With  the  triangular  diagram  one  can  not  tell  whether  Meyer- 
hoffer's  theorem  holds  good  unless  the  isotherms  are  marked  ;  but  this 
is  shown  directly  in  the  diagram  of  van  der  Heide  which  is  really  a 
double  diagram.*  The  temperature  is  taken  as  one  axis  and  the 
concentrations  of  the  two  salts  in  one  hundred  units  of  solution  are 
laid  off  on  the  other  axis,  right  and  left  from  a  zero  point.  In  this 
method  there  are  two  points  for  each  nonvariant  system  and  two 
curves  for  each  monovariant  system.  This  working  in  duplicate  is 
a  disadvantage ;  but  it  is,  perhaps,  compensated  by  having  the 
temperature  as  an  axis  in  the  plane  of  the  paper. 

The  direction  of  the  temperature  changes  along  the  different 
curves  can  be  predicted,  with  a  single  exception,  from  the  theorem 
of  van  Alkemade.  As  will  be  remembered,  this  theorem  states  that 
the  temperature  rises  when  going  along  a  boundary  curve  in  the 
direction  towards  the  line  connecting  the  melting  points  of  the  two 
solid  phases  in  the  system.  The  exception  is  the  line  AH.  The 
line  connecting  the  melting  points  is  the  left  side  of  the  triangle  and 
the  temperature  of  the  point  H  should  be  higher  than  that  of  A  ;  but 
this  is  not  the  case.  For  AC,  AB,  BC,  CK,  CD,  DIv,  DE,  EF  and 
BD  the  theorem  applies.  It  is  to  be  noticed  that  XX  is  the  line  con- 
necting the  melting  points  of  the  two  hydrated  double  salts  so  that  it 
is  entirely  proper  that  the  temperature  of  the  point  E  should  be 
higher  than  that  of  the  point  D. 

Only  one  isotherm  is  represented,  that  for  85°.     The  line  YY 

'Zeit.  phys.  Chem.  5,  122  (1899). 
-Ibid,  rz,  425  (1893, 


Three  Components  171 

shows  its  course,  but  merely  approximately  as  there  are  not  sufficient 
data  to  permit  of  its  being  drawn  accurately  to  scale.  It  is  composed 
of  four  parts  while  the  isotherm  for  72®  and  92°  have  three  parts 
only.  At  temperatures  between  47.2°  and  48.2°  the  isotherm  will 
be  made  up  of  four  sections  because  it  cuts  the  lines  KC,  CD  and 
EB.  Since  the  lowest  temperature  at  which  a  solution  can  exist  is 
the  point  H  in  the  side  of  the  triangle,  there  will  be  no  isotherm 
which  will  form  a  closed  curve.  At  — 4.5°  the  isotherm  will  consist 
of  two  lines  from  the  point  A  meeting  the  side  of  the  triangle,  one 
above  and  the  other  below  the  point  H.  With  falling  temperature 
the  intersection  of  the  two  lines  will  pass  along  the  curve  AH  until 
at  H  the  isotherm  becomes  a  point.  The  data  for  the  system  com- 
posed of  magnesium  snlfate,  potassium  snlfate  and  water  are  given 
in  Table  XXVI.  The  concentrations  are  reacting  weights  in  one 
hundred  reacting  weights  of  solution,  x  referring  to  potassium  sul- 
fate  and  y  to  magnesium  sulfate. 

TABUS  XXVI 
Temp.        -r          jr        Temp.        .v          jr 


Curve  BE 

Curve  AB 

-    •  : 

0.89 

i-37 

-  4.5°      0.87       1.54 

+  10. 

I.I7 

1-63 

Curve  BC 

20. 

i-39 

1.84 

+22.          1.24      3.78 

30- 

1-63 

2.07 

47-3        J-53      5-75 

40. 

i.8y 

Curve  CD 

2-05 

2.76 

72.           1.69      6.12 

60. 

2.17 

3-" 

Curve  DL 

70. 

2.25 

3-24 

r  =            1.78      6.46 

80. 

2-47 

3-60 

Curve  EF 

00. 

2.58 

365 

9»           2.73      3.78 

92. 

2.67 

3-79 

In  the  system  just  studied,  one  of  the  hydrated  double  salts  was 
certainly  decomposed  by  water  at  all  temperatures  and  the  other  may 
have  been.  In  the  system  consisting  of  sodium  sulfate,  magnesium 
snlfate  and  water,  there  is  formed  a  hydrated  double  salt  which  is 
not  decomposed  by  water  at  certain  temperatures.  This  equilibrium 


172 


The  Phase  Rule 


has  been  studied  to  a  certain  extent  both  by  van  't  Hoff l  and  by 
Roozeboom.1  The  solubility  determinations  are  represented  graph- 
ically in  Fig.  42,  the  concentrations  being  reacting  weights  in  one 


FIG.  42. 

hundred  reacting  weights  of  the  solution.  H  and  G  are  the  cryo- 
hydric  points  for  magnesium  sulfate  heptahydrate  and  sodium  sul- 
fate  decahydrate  respectively,  the  temperatures  being — 6°  and — 0.7°. 
At  B  the  solid  phases  are  the  two  single  salts  and  ice.  The  temper- 
ature at  which  this  nonvariant  system  exists  is  unknown  as  well  as 
the  concentration  of  the  solution.  For  this  reason  the  three  curves 
meeting  at  this  point  are  represented  by  dotted  lines.  Along  BD  the 
solid  phases  are  the  two  hydrated  salts.  No  points  on  this  curve 
have  been  determined  below  E,  15°.  At  D,  21. 5°  there  appears  the 


>Zeit.  phys.  Chem.  I,  165  (1887). 

2  Ibid.  2,518  (1888). 


Three  Components  173 

hydrated  double  salt  corresponding  to  the  formula  Na,Mg(SO4), 
4HSO.  Along  FPF,  the  solid  phases  are  magnesium  sulfate  hepta- 
hydrate  and  the  hydrated  double  salt.  In  the  direction  PF  this 
curve  will  be  terminated  by  the  appearance  of  the  hexahydrate  as 
solid  phase.  The  co-ordinates  of  this  new  nonvariant  system  have 
not  been  determined.  The  part  of  the  curve  PF,  represents  a  labile 
equilibrium,  instable  with  respect  to  the  decahydrate  of  sodium  sul- 
fate. Along  the  curve  DC  the  solid  phases  are  the  hydrated  double 
salt  and  hydrated  sodium  sulfate.  At  C,  30°,  there  appears  a  new 
solid  phase,  namely  anhydrous  sodium  sulfate.1  Along  CK  the  solid 
phases  are  anhydrous  sodium  sulfate  and  the  hydrated  double  salt. 
This  curve  has  only  been  followed  a  little  way  and  it  is  not  known 
what  becomes  of  it  at  higher  temperatures.  It  is  rather  important 
that  this  should  be  investigated  since  it  is  impossible  to  predict 
whether  the  curve  will  bend  upwards  or  downwards.  Its  present  di- 
rection is  contrary  to  the  theorem  of  van  Alkemade  ;  but  we  are  not 
justified  in  calling  it  a  real  exception  until  the  whole  curve  has  been 
determined.  From  the  point  C  there  must  run  a  third  curve  along 
which  anhydrous  and  hydrated  sodium  sulfate  are  solid  phases  ;  but 
the  only  other  point  on  the  curve  which  is  known  is  at  N,  32.6°,  for 
the  binary  system  composed  of  sodium  sulfate  and  water.  For  this 
reason  a  dotted  line  has  been  drawn  between  these  two  points  to  em- 
phasize the  fact  that  they  are  connected.  The  experimental  curve 
will  probably  have  this  general  form. 

The  fields  for  the  divariant  systems,  in  which  there  is  only  one 
solid  phase,  are  marked  on  the  diagram  so  far  as  possible.  Ice  ex- 
ists in  the  field  bounded  by  GBH  ;  magnesium  sulfate  heptahydrate 
in  the  field  bounded  by  HBDF  and  the  undetermined  line  for  the 
hexahydrate.  The  field  for  sodium  sulfate  with  ten  of  water  is 
bounded  by  GBDCN.  The  field  for  anhydrous  sodium  sulfate  can 
not  be  given  until  something  definite  is  known  about  the  curve  CK. 
It  will  be  possible  to  have  two  different  divariant  s)-stems  in  the 
field  between  CK  and  CN  and  also  in  the  field  between  CK  and  CD. 


1  The  observation  of  van  't  Hoff  that  the  hydrated  sodium  snlfate  melts  at 
26°  in  presence  of  an  equivalent  quantity  of  crystallized  magnesium  sulfate  can 
not  refer  to  a  state  of  stable  equilibrium  and  has  no  place  in  this  discussion. 


174  The  Phase  Rule 

In  the  field  KCN  one  of  the  solid  phases  will  be  hydrated  sodium 
sulfate  while  the  solid  phase  of  the  second  divariant  system  will  be 
anhydrous  sodium  sulfate  or  the  hydrated  double  salt  according  as 
CK  curves  in  one  direction  or  the  other.  Although  no  measure- 
ments have  been  made  in  regard  to  this,  Roozeboom  feels  certain 
that  anhydrous  sodium  sulfate  exists  as  the  solid  phase  to  the  right 
of  CK  and  the  hydrated  double  salt  to  the  left.1  It  is,  of  course, 
clear  that  two  sets  of  divariant  systems  can  not  exist  at  the  same 
temperature  with  the  same  concentrations  in  the  solution  phase.  We 
can,  therefore,  have  different  isotherms  apparently  intersecting  in 
the  fields  KCN  and  KCD.  This  indefiniteness  disappears  if  we  con- 
sider the  solid  figure  formed  with  the  temperature  as  vertical  axis. 
The  reason  for  the  complication  is  that  the  solubility  of  anhydrous 
sodium  sulfate  decreases  with  increasing  temperature. 

While  we  do  not  know  the  whole  of  the  field  for  the  hydrated 
double  salt,  it  is  clear  from  the  diagram  that  this  compound  can  ex- 
ist as  solid  phase  in  the  field  to  the  right  of  the  lines  FDC.  The 
dotted  lines  XX  represent  solutions  which  contain  the  two  salts  in 
equivalent  quantities.  This  line  cuts  the  curve  DC  at  the  point  X, 
corresponding  to  a  temperature  of  25°,  showing  that  the  hydrated 
double  salt  is  decomposed  by  water  at  a  temperature  below  25°  while, 
from  this  temperature  on,  water  will  dissolve  the  solid  components 
in  equivalent  quantities.  This  is  found  to  be  the  case  experiment- 
ally. Above  25°  the  hydrated  double  salt  is  not  decomposed  by 
water  ;  but  dissolves  and  recrystallizes  as  if  it  were  a  single  sub- 
stance. Below  21.5°  the  double  salt  cannot  exist  at  all  and 
between  21.5°  and  25°  it  can  exist  only  in  presence  of  a  solu- 
tion containing  a  relatively  larger  amount  of  magnesium  sulfate  than 
the  crystals.  In  other  words  between  these  two  temperatures  the 
hydrated  double  salt  is  decomposed  by  water  with  precipitation  of 
hydrated  sodium  sulfate.  The  temperatures  between  the  limits  within 
which  the  double  salt  is  decomposed  have  been  called  the  range  of 
decomposition  by  Meyerhoffer.2  For  this  particular  compound  the 
range  is  only  about  three  and  a  half  degrees  ;  for  the  double  sulfate 


1  Private  letter. 

'Zeit.  phys.  Cheni.  5,  109  (1890). 


Three  Components  175 

of  magnesium  and  potassium  with  six  of  water  we  have  seen  that  it 
covered  all  the  temperatures  at  which  that  salt  could  exist.  With 
the  double  sulfate  of  copper  and  potassium  the  range  of  decomposi- 
tion is  presumably  zero  though  there  are  no  experiments  to  that  ef- 
fect. Within  the  range  of  decomposition  a  compound  can  not  be 
purified  by  recrystallization  because  it  is  continually  decomposed  by 
water.  In  Table  XXVII  are  the  data  for  the  system,  magnesium 
and  sodium  sulfates  and  water.  The  concentrations  are  expressed 
as  reacting  weights  of  the  salts  in  one  hundred  reacting  weights  of 
the  solution,  x  denoting  sodium  sulfate  and  y  magnesium  sulfate. 

TABLE  XXVII 
Temp.         x          y         Temp.         x          y 


Curve  BED 

Curve  F,DF 

15.°         1.55      4.25 

18. 

5°        3-i7      3-96 

18.5       ;    2.01        4.29 

22. 

2.66      4.32 

Curve  DC 

24. 

5      ;    2.51       4.44 

22.            2.74      4.37 

30- 

2-14      4-93 

24-5         3-23      3-39 

35- 

1.61       5.46 

30.        i   4.27      2.71 

Curve  CK 

35- 

\   4.02  |   2.58 

The  equilibrium  between  cupric  chloride,  potassium  chloride  and 
water  has  been  studied  by  Meyerhoffer1  though  his  investigations 
cover  only  a  small  portion  of  the  field.  His  results  are  represented 
graphically  in  Fig.  43.  G  is  the  cryohydric  point  for  ice  and  potas- 
sium chloride,  the  temperature  being  — 11.4°  according  to  Guthrie*. 
H  is  the  corresponding  point  for  cupric  chloride  with  two  of  water. 
This  point  has  not  been  accurately  determined  but  the  temperature 
is  lower  than  — 23°  3.  Along  GB  ice  and  potassium  chloride  are 
solid  phases  and  along  HX  ice  and  hydrated  cupric  chloride.  These 
curves  do  not  meet,  as  in  all  probability  the  double  chloride  with  the 
composition  corresponding  to  the  formula  CuCl^zKCl2H,O  appears 
both  at  B  and  at  N.  Each  of  these  points  represents  a  nonvariant 


1  Zeit.  phys.  Chem.  3,  336  ( 1889). 

-  Phil.  Mag.  (4)  49.  269  (1875). 

*Cf.  de  Coppet,  Ann.  chim.  phys.  (4)  «£,  386  (1871). 


1 76 


The  Phase  Rule 


system,  the  solid  phases  at  B  being  ice,  potassium  chloride  and 
copper  dipotassium  chloride,  while  at  N  they  are  ice,  hydrated  cupric 
chloride  and  the  hydrated  double  salt.  Along  BN  the  solid  phases 
are  ice  and  the  hydrated  double  salt.  Not  one  of  these  curves  has 
been  studied  experimentally  ;  but  the  temperature  should  rise  in 
passing  from  N  to  B.  Along  BD  the  solid  phases  are  potassium 
chloride  and  the  blue  copper  dipotassium  chloride.  This  curve  has 
been  followed  from  E,  39°,  to  D,  92°.  At  this  latter  temperature 


FIG.  43. 

there  is  formation  of  the  red  anhydrous  double  salt  represented  by 
the  formula  CuCl2KCl.  At  D  there  exists  the  non variant  system, 
potassium  chloride,  copper  dipotassium  chloride,  copper  potassium 
chloride,  solution  and  vapor.  If  the  temperature  rises  above  92° 
the  copper  dipotassium  chloride  breaks  up  into  potassium  chloride 
and  copper  potassium  chloride  forming  the  monovariant  system 
stable  along  DF,  the  two  last  mentioned  salts  being  present  as  solid 


Three  Components  :  -  - 

phases.  Going  back  and  starting  from  X  we  can  pass  along  the 
curve  XC  with  hydrated  cnpric  chloride  and  the  bine  ternary  com- 
pound as  solid  phases.  The  point  S  corresponds  to  the  temperature 
of  zero  degrees  centigrade  and  is  the  point  at  which  Meyerbofier  s 
measurements  begin.  Beyond  C,  56°,  the  copper  dipotassium 
chloride  ceases  to  be  stable  in  presence  of  an  excess  of  cnpric 
chloride  and,  at  this  point,  there  exists  the  nonvariant  system,  hy- 
drated capric  chloride,  copper  dipotasaum  chloride,  copper  potassmn 
chloride,  solution  and  vapor.  Along  CK  the  solid  phases  are  hy- 
drated cnpric  chloride  and  the  anhydrous  doable  salt  while  along  CD 
they  are  the  ternary  and  the  binary  compounds  of  copper  and  potas- 
sium chloride. 

Ice  exists  as  solid  phase  in  the  field  bounded  by  HXBG  ; 
potassium  chloride  in  that  bounded  by  GBDF  ;  and  hydrated  cnpric 
chloride  in  the  field  bounded  by  HXCK  and  the  nndetermined  curve 
for  the  anhydrous  salt.  The  blue  hydrated  doable  salt  exists  as 
solid  phase  only  in  the  closed  field  DBXCD.  The  dotted  line  ZZ 
represents  solutions  in  which  there  is  twice  as  much  potassium 
chloride  as  cnpric  chloride.  This  line  does  not  cot  the  field  for 
copper  dipotassium  chloride  so  far  as  we  know  and  therefore  this 
salt  is  always  decomposed  by  water  with  precipitation  of  potassium 
chloride.  The  green  color  which  this  salt  usually  has  is  due  to  a 
slight  decomposition  and  traces  of  mother  liquor  rich  in  cnpric 
chloride.  To  obtain  the  salt  pure  it  should  be  washed  with  a  potas- 
sium chloride  solution  instead  of  with  pore  water.  The  dotted  line 
XX  represents  solutions  in  which  the  ratio  of  cnpric  and  potassium 
chlorides  is  unity.  The  field  for  the  anhydrous  doable  salt,  bounded 
by  KCDF,  is  cut  by  this  line  at  X,,  72°,  and  from  tins  temperature 
upwards  it  is  possible  to  obtain  a  solution  of  this  doable  salt  contain- 
ing the  two  salts  in  the  same  ratio  as  in  the  solid  phase.  Between 
56°  and  72°  the  red  double  salt  is  decomposed  by  water  with  forma- 
tion of  the  bine  salt  CuCl,  aKCLzH^O.  In  this  case  the  range  of  de- 
composition extends  over  sixteen  degrees. 

The  line  ZZ  connects  the  melting  points  of  ice  and  of  copper  di- 
potassium chloride  and  it  is  for  this  reason  that  the  temperature 
should  rise  as  the  system  passes  along  XB  from  X  to  B.  It  is  to  be 
noticed  that  the  salt  component  which  is  present  at  B  is  necessarily 
12 


178  The  Phase  Rule 

the  component  which  is  precipitated  by  the  action  of  water  upon  the 
double  salt.  This  has  been  expressed  in  another  form  by  Schreine- 
makers1  in  the  rule  :  ' '  The  cryohydric  temperature  of  a  solution  in 
equilibrium  with  double  salt  and  the  component  which  does  not  pre- 
cipitate is  lower  than  the  cryohydric  temperature  of  a  solution  in 
equilibrium  with  double  salt  and  the  component  which  precipitates." 
If  the  double  salt  were  not  decomposed  by  water,  the  points  N  and 
B  in  the  diagram  would  lie  on  opposite  sides  of  the  line  ZZ  and 
there  would  be,  according  to  the  theorem  of  van  Alkemade,  a  maxi- 
mum temperature  at  the  point  where  ZZ  cuts  NB.  This  conclusion 
was  drawn  simultaneously  by  Schreinemakers2  and  by  Meyerhoffer,3 
who  reached  this  result  independently  and  by  different  ways.  As 
an  illustration  of  this  point  we  may  take  the  system,  copper  sulfate, 
ammonium  sulfate  and  water.  The  temperature  at  which  ice,  copper 
sulfate  with  five  of  water  and  the  hydrated  double  salt  Cu(NH4)2 
(SO4),6H,O  are  in  equilibrium  with  solution  and  vapor  is  —2.6°. 
The  temperature  for  the  corresponding  point  with  ammonium 
sulfate  instead  of  copper  sulfate  as  solid  phase  is  —19°.  A  solution 
containing  the  two  salts  in  equivalent  quantities  is  in  equilibrium 
\vith  ice  and  the  double  salt  at  —  i .  7° .  The  double  salt  of  copper  and 
ammonium  chloride  CuCl22NH4Cl2H2O  is  in  equilibrium  with  ice  and 
ammonium  chloride  at  — 15.7°  ;  with  ice  and  hydrated  cupric 
chloride  at  a  temperature  lower  than  —42°,  while  the  curve  connect- 
ing these  points  passes  through  a  maximum  temperature  at  —12.7°. 

In  table  XXVIII  are  the  data  for  the  system,  potassium  chloride, 
cupric  chloride  and  water,  x  denoting  reacting  weights  of  potassium 
chloride,  y  reacting  weights  of  cupric  chloride  in  one  hundred  react- 
ing weights  of  solution. 

The  system,  lead  iodide,  potassium  iodide  and  water  studied  by 
Schreinemakers4  need  not  detain  us  long.  There  is  formed  a  com- 


'  Zeit.  phys.  Chem.  12,  85  (1893). 

*Ibid.  12,  87  (1893), 

*Sitznngsber.  Akad.  Wiss.  Wien,  1  O2,  lib,  2co  (1893). 

4  Zeit.  phys.  Chem.  9,  57  ;  io>  467  ( 1892)  ;  Schreinemakers  was  misled  by 
a  faulty  analysis  made  by  Ditte  and,  in  the  first  paper,  the  formula  of  the  hy- 
drated double  salt  is  wrong  and  consequently  most  of  the  theoretical  conclu- 
sions. In  the  secoikl  paper  the  errors  are  corrected, 


Three  Components  ,79 

pound,  PbIsKl2H,O,  which  is  decomposed  at  all  temperatures  with 
precipitation  of  hydrated  lead  iodide.  The  cryohydric  temperature 
for  double  salt  and  lead  iodide  is  —2.8°  and  for  double  salt  and 
potassium  iodide  is  —22.8°.  This  is  an  experimental  confirmation 
of  the  rule  of  Schreinemakers  in  regard  to  the  relative  temperatures 
of  these  points.  The  hydrated  double  salt  is  an  excellent  one  for  a 
lecture  experiment  since  it  is  pale  yellow  in  color,  while  the  hydrated 
lead  iodide  crystallizes  in  lustrous  almost  orange  scales.  Addition  of 
water  produce  an  almost  instantaneous  decomposition  with  change  of 
color.  This  can  be  made  to  disappear  by  adding  potassium  iodide. 

TABLE  XXVIII 


Temp. 

X 

y 

Temp. 

_r 

y 

Curve  BED 
39.4°        8.61      4.82 
49-9          9-68      5.43 
60.4        1  1.2        6.35 
79.1         14.47      8-54  ij 
90.5         17.5       10.69 
Curve  DF 

93-7      |  i»-3      "-4 
98.8        20.1       12.3 

Curve  XSC 
o.°         1.7 
396         4-6 
50-i         57 
52.9         6.25 
Curve  CK 
60.2         6.84 
72.6         6.6S 
Curve  CD 
64.2         9.16  ;j 

+ 

8.81 
10.9 
11.38 
12.08 

12.74 
13-43 

11.78 

There  are  no  other  ternary  systems  made  up  of  two  salts  and 
water  which  have  been  studied  at  all  in  detail ;  but  there  are  deter- 
minations of  several  quintuple  points  or  points  at  which  five  phases 
coexist.1  Calcium  acetate  combines  with  copper  acetate  to  form  a 
double  salt2  which  splits  into  the  component  salts  at  76°.  The  solid 
phases  at  this  point  are  CaACjHjO,  CuAcjHsO  and  CaCuAc.SH/X 
This  is  a  striking  experiment  because  copper  acetate  is  green  and 
calcium  acetate  white,  while  the  hydrated  double  salt  is  an  intense 
blue.  In  this  case  the  double  salt  is  stable  below  the  inversion  tem- 
perature while  the  double  sulfate  of  magnesium  and  sodium  was 
stable  at  higher  temperatures,  decomposing  into  single  salts  at  tem- 


1  Roozeboom,  Recnefl  Trav.  Pajs-Bas  6,  331,  (1887)  ;  Zeit.  phja.  Chem.  a, 
513,  (1888).   . 

1  Reicher,  Ibid,  x,  221,  (1887). 


i8o  The  Phase  Rule 

peratures  below  that  of  the  quintuple  point.  The  Theorem  of  Le 
Chatelier  would  tell  us  the  direction  of  the  change  at  the  quintuple 
point  if  the  heats  of  reaction  were  known.  As  this  is  generall)7  not 
the  case,  we  must  content  ourselves  with  an  approximation.  If  we 
assume  that  no  one  of  the  compounds  has  a  true  melting  point,  the 
following  rule  holds  good  in  all  the  instances  yet  studied.  When 
one  of  the  solid  phases  can  split  into  the  other  two  with  addition  or 
subtraction  of  water1  the  inversion  point  is  a  minimum  temperature 
for  that  phase  if  the  water  be  added  to  complete  the  reaction  and  a 
maximum  temperature  if  the  water  be  subtracted.2 

The  following  instances  will  illustrate  this  rule  and  at  the  same 
time  furnish  a  list  of  the  different  systems  in  which  quintuple  points 
have  been  determined.  At  — 3°  one  of  the  double  sulfates  of  mag- 
nesium and  potassium  changes  into  the  single  sulfates  with  addition 
of  water.3  This  is  a  minimum  temperature  for  K2Mg(SO4)26H2O. 

K,Mg(S04)2  6H20  +  H20  =  K2SO4  +  MgSO4  7H2O. 
The  same  change  takes  place  at  21.5°  with  the  double  sulfate  of  so- 
dium  and    magnesium.4      This   is   the   minimum    temperature   for 
the  double  sulfate,  Na2Mg(SO4)2  4H2O. 

Na2Mg(SO4)2  4H2O  +  i3H2O  =  Na2SO4  ioH2O  -f  MgSO4  7H2O. 
Copper  potassium  chloride  changes  at  56°  into  copper  dipotassium 
chloride  and  hydrated  cupric  chloride.5     This  is  a  minimum  temper- 
ature for  CuCl,KCl. 

2CuCl2  KC1  -f  4H2O  =  CuCl2  2KC1  2H2O  -f  CuCl2  2H2O. 
At  92°  copper  dipotassium   chloride  changes  into  copper  potassium 
chloride  and  potassium  chloride.     This  is  a  maximum  temperature 
for  CuCl22KCl2H2O. 

CuCl,  2KC1  2H2O  —  2H2O  =  CuCl2  KC1  +  KC1. 


1  It  is  not  possible  to  have  in  equilibrium  three  solid  phases  such  that  one 
can  be  formed  from  the  other  two  without  addition  or  subtraction  of  water. 

2Cf.  Roozeboom,  Recueil  Trav-Pays-Bas,  6,  341  (1887)  ;  Zeit.  phys.  Chem. 
«,  517  (1888),  Bancroft,  Jour.  Phys.  Chem.  I,  No.  6  (1897). 

3  van  der  Heide,  Zeit.  phys.  Chem.  J2,  416  (1893). 

4  van  't  HofF  and  van  Deventer,  Ibid.  I,  165  (1887). 
'*  Meyerhoffer,  Ibid.  3,  336  (1889). 


Three  Components  181 

An  analogous  change  occurs  at  146°  with  copper  diammoniutn  chlo- 
ride.1    This  is  a  maximum  temperature  for  CuCl,  2XH4Cl2H,O. 

Cud,  2XHtCl2H±O  — 2HtO  =  CuCltXH4Cl  +  NH4Cl. 
At    15.5°    one  of  the  double  salts  of  copper  tetrethyl   ammonium 
chloride  changes  into  another  double  salt  and  hydrated  cupric  chlo- 
ride.1    This  is  a  minimum  temperature  for  5CuClt  2X(C,H$)4Cl. 
5CUC1,  2N(C,Hi)tCl  +  8H,O  =  CuClt  2X(C1H5)4C1  +  4CuClt  zI^O. 
The  double  acetate  of  copper  and  calcium  changes  at  76°  into  the 
single  acetates.3     This  is  a  maximum  temperature  for  CuCaAc4  8H,O. 

CuCaAct  8H.O  —  6H.O  =  CuAc,  H,O  +  CaAc.  H.O. 
Sodium  ammonium  racemate  decomposes  at  27°  into  the  dextrorotary 
and  laevorotary  sodium  ammonium  tartrates.*     This  is  a  minimum 
temperature  for  (NaNH4C4H4O€H,O)t. 

(NaNHtCtHtOt  H,O)t  +  6HZO  =  2(NaNH4C4H4Os4HiO). 
The  same  salt  changes  at  35°  into  the  single  racemates.5     This  is  a 
maximum  temperature  for  (NaNH4C4H4O4HjO),. 

2(NaNH4CtH408HJ0)1-4HiO=(NaIC4H40<)t-f-([NH4]1C4H4Os)2. 
The  double  potassium  sodium  racemate  undergoes  similar  changes  at 
the  temperature  of  —6°  and  41°  respectively*  the  first  being  a  mini- 
mum temperature  for  (KXaCjH^O^H.O),  and  the  second  a  maxi- 
mum temperature  for  the  same  salt. 

(KNaC4H4Os  3HtO)J-f-2H1O=2(KXaC4H4Os  4H,O). 
2(KNaC4H4Os  3H,0)t  -  8H,O  =  (Na,C4HtO.\  -f  K,C4H4OS  2H,O), . 
There  are  some  nonvariant  sj^stems  containing  three  solid  phases, 
solution  and  vapor,  in  which  two  of  the  solid  phases  can  not  be 
made  from  the  third  with  or  without  addition  of  water  and  these 
must  also  be  considered.  They  can  be  divided  into  two  classes  : 
' '  One  solid  phase  can  be  transformed  into  one  of  the  others  by  ad- 


Meyerhoffer,  ZeiL  phys.  Chern.  5,  98  (1890). 

Meyerhoffer,  Sitzungsber.  Akad.  Wiss.  Wien,  TOJB,  lib,  150  (1893). 

Reicher,  Zeit  ph3'S.  Chem.  I,  221  (1887). 

vati  'tHoff  aud  van  Deventer,  Ibid.  I,  165  (1887). 

van  't  HoflF,  Goldschmidt  and  Torisseu,  Ibid.  17,  49  (1895). 

van  *t  Hoff  and  Goldschmidt,  Ibid.  17,  505  ( 1895). 


1 82  The  Phase  Rule 

dition  or  subtraction  of  water.  No  one  of  the  solid  phases  can  be 
converted  into  either  of  the  others  by  addition  or  subtraction  of 
water."  If  one  solid  phase  can  be  converted  into  one  of  the  others 
by  addition  of  water  the  inversion  point  is  a  maximum  or  a  mini- 
mum temperature  for  one  of  those  phases  and  is  neither  a  maximum 
nor  minimum  for  the  the  third  phase.  This  can  be  illustrated  very 
readily  by  three  instances  from  the  system,  potassium  sulfate,  mag- 
nesium sulfate  and  water.  At  47.2°  two  of  the  solid  phases  are 
MgSO4  7H2O  and  MgSO,  6H,O  while  the  third  is  K2Mg(SO4)26H2O. 
This  is  a  minimum  temperature  for  the  hexahydrate.  If  the  tem- 
ature  of  the  point  had  been  higher  than  48.2°  it  would  have  been  a 
maximum  temperature  for  the  heptahydrate.  No  example  of  this 
latter  form  has  yet  been  observed  ;  but  it  is  not  impossible.  The 
hydrated  double  salt  exists  both  above  and  below  the  temperature  of 
the  inversion  point.  At  72°  two  of  the  solid  phases  are  the  hydrated 
double  salts  K2Mg(SO4)2  6H2O  and  K2Mg(SO,)24H2O  while  the 
third  is  magnesium  sulfate  heptahydrate.  At  92°  two  of  the  solid 
phases  are  the  same  two  hydrated  double  salts  and  the  third  is  po- 
tassium sulfate.  The  first  temperature  is  a  minimum  for  the  double 
salt  with  four  of  water  and  the  second  a  maximum  for  that  with  six 
of  water.  If  the  temperatures  are  not  given  it  can  only  be  told  by 
experiment  which  point  is  which.  If  the  compositions  of  the  solu- 
tions are  known,  the  direction  of  the  temperature  change  can  be 
foretold  from  the  theorem  of  van  Alkemade.  The  higher  tempera- 
ture will  necessarily  be  a  maximum  for  the  double  salt  with  the  larger 
amount  of  water  of  crystallization. 

When  no  one  of  the  three  solid  phases  can  be  converted  into  either 
of  the  others,  it  is  impossible  to  make  any  definite  prediction  when 
the  only  data  are  the  formulas  of  the  three  solid  phases.  As  an  in- 
stance take  the  two  quintuple  points  where  the  solid  phases  are  ice, 
hydrated  calcium  acetate  and  copper  calcium  acetate,  ice,  hy- 
drated copper  acetate  and  copper  calcium  acetate.  The  two  sets 
consist  of  ice,  a  hydrated  salt  and  a  hydrated  double  salt.  There  is 
no  way  of  distinguishing  them  without  further  information.  Here 
again  the  theorem  of  van  Alkemade  will  help  us  if  the  concentra- 
tions of  the  two  solutions  are  known  and  if  the  double  salt  is  de- 
composed by  water.  If  the  double  salt  is  stable  in  presence  of  water, 


Three  Components  183 

there  is  no  a  priori  method  of  telling  which  cryohydric  tempreature 
is  the  higher,  though  this  may  be  guessed  at  if  the  concentrations 
are  known.1  It  should  be  clearly  understood  that  in  all  these  cases 
a  maximum  or  a  minimum  temperature  for  a  given  substance  refers 
to  that  substance  in  equilibrium  with  solution  and  vapor.  For  in-  • 
stance,  56°  is  a  minimum  temperature  for  copper  potassium  chloride 
in  equilibrium  with  solution  and  vapor  ;  but  it  is  possible  for  copper 
dipotassium  chloride,  copper  potassium  chloride,  potassium  chloride 
and  vapor  to  be  in  stable  equilibrium  at  ordinary  temperatures. 

It  is  not  an  entirely  empirical  rule  that  when  one  solid  phase 
can  change  into  the  other  two  with  addition  or  subtraction  of  water 
the  quintuple  point  is  a  minimum  or  a  maximum  temperature  res- 
pectively for  that  phase.  The  heat  effect  due  to  the  absorption  or 
splitting'  off  of  water  is  so  much  greater  than  any  of  the  other  heats 
of  reaction  that  it  determines  the  sign  of  the  whole  heat  effect  and 
the  direction  of  the  change.  In  cases  where  this  is  not  so,  the  rule 
will  not  hold.  One  would  expect  it  to  apply  in  practically  all  cases 
where  two  of  the  components  are  solids  in  the  pure  state  at  all  tem- 
peratures covered  by  the  experiment,  and  where  the  third  component 
is  near  or  above  its  melting  point.  It  so  happens  that  most  systems 
which  have  been  studied  come  under  this  head.  Water  occupies  an 
exceptional  position  in  determining  the  direction  of  the  change  be- 
cause it  occupies  an  exceptional  place  in  the  system,  being  the  sol- 
vent and  practically  the  only  constituent  in  the  vapor  phase.  It  is 
obvious  that  it  is  necessary  to  limit  these  rules  for  the  change  of  the 
equilibrium  with  the  temperature  to  compounds  not  having  a  true 
melting  point,  because  the  melting  point  is  always  the  maximum 
temperature  for  that  phase,  and  neither  of  the  two  quintuple  points 
in  which  a  boundary  curve  terminates  is  necessarily  a  maximum 
temperature  even  for  the  monovariant  S3*stem  which  exists  along  it. 


Cf.  Schreinemakers,  Zeit  phys.  Chem.  I«,  73  (1893). 


CHAPTER  XIII 


PRESSURE    CURVES 

All  the  monovariant  systems  considered  thus  far  have  been  com- 
posed of  two  solid  phases,  solution  and  vapor,  for  only  these  find  a 
place  in  a  concentration-temperature  diagram.  In  the  quintuple  point 
five  boundary  curves  meet  so  that  there  are  still  two  monovariant 
systems  unaccounted  for.  Instead  of  taking  these  by  themselves,  it 
will  be  better  to  treat  them  as  parts  of  the  pressure-temperature  dia- 
gram for  a  system  of  three  components,  only  one  of  which  is  meas- 
urably volatile.  In  the  simplest  case  that  we  need  consider,  one  of 
the  salts  forms  a  hydrate,  which  can  coexist  as  solid  phase  with  any 
of  the  three  other  possible  solid  phases,  and  is  instable  at  its  melting 
point.  The  concentration-temperature  diagram  for  this  system  is 
given  by  I  a  in  Fig.  36.  Such  a  system  would  be  realized  with  sodi- 
um sulfate,  sodium  chloride  and  water,  denoted  by  C,  B  and  A,  re- 
spectively. Fig.  44  is  the  pressure-temperature  diagram  for  the  equi- 
libria around  the  quintuple  point  K  where  sodium  chloride,  hydrated 
and  anhydrous  sodium  sulfate,  solution  and  vapor  coexist.  KF,  KE 


FIG.  44. 

and  KO  are  the  curves  for  the  monovariant  systems  composed  of  two 
solid  phases,  solution  and  vapor.  The  solid  phases  are  hydrated  and 
anhydrous  sodium  sulfate  along  KE,  anhydrous  sodium  sulfate  and 


Three  Components  185 

sodium  chloride  along  KF,  hydrated  sodium  sulfate  and  sodium  chlo- 
ride along  KO.  At  the  point  E  the  concentration  of  sodium  chloride 
is  zero,  and  the  concentration  of  water  is  zero  at  the  point  F.  The 
curve  KO  terminates  at  the  point  O  where  ice  appears  as  a  solid 
phase,  making  a  new  quintuple  point.  The  solution  connoted  by 
KE  is  more  dilute  than  the  one  connoted  by  KF  and  therefore  the 
former  curve  lies  above  the  latter.  The  curves  KE  and  KF,  if  pro- 
longed, will  lie  below  KO.  When  one  of  the  phases  is  a  saturated 
solution,  the  more  stable  system  is  the  one  with  the  higher  vapor 
pressure.1  This  is  true  for  ternary  as  well  as  for  binary  systems.  It 
does  not  follow  from  this  that  the  monovariant  SN'stem  represented 
by  KF  is  instable  with  respect  to  that  represented  by  KE.  It  is  not 
sufficient  to  consider  the  total  concentrations  only  ;  with  three  com- 
ponents the  relative  amounts  of  the  two  salts  is  a  very  important 
factor  in  determining  the  equilibrium.  If  the  solution  with  the  lower 
vapor  pressures  along  KF  could  change  into  the  solutions  with  the 
higher  vapor  pressures  along  KE  by  precipitation  of  hydrated  sodium 
sulfate,  the  first  set  of  solutions  would  not  represent  stable  modifica- 
tions. This  is  not  the  case,  and  both  KF  and  KE  are  pressure-tem- 
perature curves  for  stable  monovariant  systems. 

Of  the  other  two  curves  meeting  in  the  point  K,  the  curve  KN 
represents  the  conditions  of  pressure  and  temperature  under  which 
hydrated  and  anhydrous  sodium  sulfate,  sodium  chloride  and  solu- 
tion are  in  equilibrium.  According  to  the  theorem  of  Le  Chatelier, 
this  curve  will  slant  to  the  left  if  hydrated  sodium  sulfate  and  sodi- 
um chloride  occupv  a  larger  volume  than  anhydrous  sodium  sulfate 
and  the  resulting  solution  ;  to  the  right  if  the  reverse  is  the  case. 
The  curve  KS  represents  the  equilibrium  between  hydrated  and  an- 
hydrous sodium  sulfate,  sodium  chloride  and  vapor.  This  curve,  if 
prolonged,  would  certainly  lie  above  KF  and  possibly  above  KE  be- 
cause the  system  with  the  higher  vapor  pressure  is  the  less  stable 
when  it  contains  no  solution  phase. 

The  quantitative  relations  of  EK  and  KS  to  the  boundary  curves 
for  the  binary  system,  sodium  sulfate  and  water,  are  not  definitely 
known.  In  Fig.  45  are  given  the  two  chief  possibilities.  The  quad- 


1  See  page  58. 


The  Phase  Rule 


lf\ 


FIG.  45. 

ruple  point  for  the  binary  system  is  at  E  in  both  diagrams  and  the 
quintuple  point  for  the  ternary  system  at  K.  Along  curves  I,  II, 
III  and  IV  the  solid  phases  are  anhydrous  sodium  sulfate,  solution 
and  vapor  ;  hj'drated  and  anhydrous  sodium  sulfate,  and  solution  ; 
hydrated  sodium  sulfate,  solution  and  vapor ;  hydrated  and  an- 
hydrous sodium  sulfate,  and  vapor.  Curve  V  represents  the  instable 
portion  of  curve  I,  the  phases  being  anhydrous  sodium  sulfate,  solu- 
tion and  vapor.  The  curves  for  the  ternary  system  have  the  same 
lettering  as  in  Fig.  44.  In  the  left-hand  diagram  it  is  assumed  that 
sodium  sulfate  effloresces  under  water  when  the  pressure  equals  the 
dissociation  pressure  for  pure  hydrated  sodium  sulfate  ;  in  the  right- 
hand  diagram  it  is  assumed  that  the  change  takes  place  when  the 
pressure  equals  that  of  the  solution  in  equilibrium  with  anhydrous 
sodium  sulfate.  If  the  first  hypothesis  is  the  right  one,  EKS  be- 
comes a  single  curve  and  will  always  have  the  same  values  regardless 
of  the  nature  of  the  third  component.  This  seems  to  be  the  view 
adopted  by  Lowenherz1.  An  analogous  case  occured  in  the  equili- 
brium between  potassium  chloride  and  water.  There  it  was  seen 
that  if  the  partial  pressure  and  the  specific  action  of  the  potassium 
chloride  were  neglected,  the  pressure-temperature  curves  for  ice, 
solution  and  vapor  and  for  ice,  salt  and  vapor  were  identical  with 
the  sublimation  curve  for  ice.2 


1  Zeit.  phys.  Cheni.  18,  70 

2  Cf.  page  50. 


1894). 


Three  Components  187 

Against  this  is  to  be  set  the  fact,  noticed  with  calcium  chloride1 
that  the  dissociation  pressure  for  a  hydrated  salt  is  a  function  of  the 
dissociation  products.  A  compound  has  a  definite  dissociation 
pressure  only  with  respect  to  a  given  phase  or  set  of  phases  just  as  a 
solution  is  saturated  only  with  respect  to  a  given  phase  or  set  of 
phases.  This  appears  very  clearly  when  the  compound  is  a  hydrated 
double  salt  which  can  effloresce  in  different  ways.  It  is  therefore 
'probable,  though  not  proved,  that  EK  and  KS  are  not  parts  of  the 
same  curve  and  that  neither  coincides  with  the  dissociation  curve  of 
hydrated  and  anhydrous  sodium  sulfates.  If  the  right-hand  diagram 
in  Fig.  45  represents  the  facts  the  curve  KS  will  lie  above  curve  IV 
or  the  presence  of  solid  sodium  chloride  decreases  the  stability  of 
hydrated  sodium  sulfate  with  respect  to  the  anhj-drous  salt.  Looked 
at  in  this  way  sodium  chloride  may  be  called  a  catalytic  agent. 

The  solubility  curve  for  a  hydrated  and  anhydrous  salt  in  a 
ternary  system  has  been  studied  by  Goldschmidt.-  He  makes  the 
unnecessary  and  incorrect  assumption  that  the  solubility  in  water  of 
an  anhydrous  salt  is  scarcely  affected,  if  at  all,  by  the  addition  of  a 
non -electrolyte.  From  this  premise  he  concludes  that  addition  of  a 
non-electrolyte  increases  the  solubility  of  a  hydrated  salt.  This  is 
right  as  far  as  it  goes ;  but  it  is  inaccurately  worded  and  is  not  a 
general  statement  of  the  relations.  In  a  ternary  system  a  single 
solid  phase  has  not  a  definite  solubility  at  a  fixed  temperature.  It 
can  exist  in  equilibrium  with  a  series  of  solutions.  The  limiting 
solubility,  or  solubility  in  presence  of  another  solid  phase,  can  have 
but  a  single  value  for  each  temperature.  Goldschmidt' s  proposition 
when  accurately  worded  will  read  :  If  the  salt  concentration  in  the 
rnonovariant  system,  hydrated  and  anhydrous  salt,  solution  and 
vapor,  be  equal  to  the  solubility  of  the  anhydrous  salt  in  water,  it 
will  be  greater  than  the  solubility  of  the  hydrated  salt  in  water. 
This  is  only  a  special  case  of  the  well-known  fact  that  one  thing  can 
not  be  equal  to  two  different  things.  It  is  probable  that  the  limiting 
solubility  is  determined  in  cases  of  this  sort  by  pressure  considera- 
tions and  it  is  for  this  reason  that  Goldschmidt 's  results  have  been 


1  Cf.  page  73. 


"  \.\.  page  73. 

*Zeit.  phys.  Cbem.  17,  145  (1895). 


i88 


The  Phase  Rule 


referred  to  here  rather  than  in  the  chapters  on  concentration  and 
temperature.  It  is  very  important  that  some  careful  measurements 
be  made  to  determine  directly  the  pressure  at  which  the  efflorescence 
takes  place.  The  chief  objection  to  accepting  the  left-hand  diagram 
in  Fig.  45  is  the  behavior  of  hydrated  double  salts  ;  but  it  is  conceiv- 
able that  these  really  come  in  a  class  by  themselves  and  that  one 
might  lay  down  the  rule  that  a  solid  phase  affects  the  dissociation 
pressure  of  another  solid  phase  only  when  it  can  react  with  one  of 
the  dissociation  products. 

Of  special  interest  are  the  curves  for  the  monovariant  systems  in 
which  a  hydrated  double  salt  is  one  of  the  solid  phases.  As  an  ex- 
ample of  a  hydrated  double  salt  which  separates  from  solution  only 
below  a  certain  temperature,  we  may  take  the  system,  copper  acetate, 
calcium  acetate  and  water,  in  the  neighborhood  of  76°.  Roozeboom1 
has  already  pointed  out  the  relative  positions  of  the  five  curves  at 
this  quintuple  point  and  his  diagram  is  reproduced  in  Fig.  46. 


FIG.  46. 

OD,  OOt ,  and  OO2  are  the  curves  for  two  solid  phases,  solution  and 
vapor.  Along  OD  the  solid  phases  are  the  two  single  acetates, 
CuAc2H2O  and  CaAc2H2O  ;  along  OOt ,  the  hydrated  double  salt, 
CuCaAc4  8HZO,  and  crystallized  copper  acetate  ;  along  OO2  the  hy- 
drated double  salt  and  crystalized  calcium  acetate.  The  curve  DO  , 


'Zeit.  phys.  Chem.  2,  517  (1888). 


Three  Components  189 

if  prolonged  will  lie  below  OO4  for  the  reasons  already  given  when 
discussing  Fig.  44.  OE  is  the  curve  for  the  double  salt,  the  two 
single  salts  and  solution.  The  curve  slants  to  the  left  because  the 
hydrated  double  salt  occupies  a  larger  volume  than  its  dissociation 
products  and  must,  therefore,  be  decomposed  by  pressure.1  Spring 
and  van  't  Hoff1  have  found  that,  under  an  estimated  pressure  of 
six  thousand  atmospheres,  the  hydrated  double  salt  is  certainly  con- 
verted into  the  single  salts  and  solution  at  40°  and  probably  at  16°. 
The  curve  OF  gives  the  simultaneous  values  for  pressure  and  tem- 
perature when  the  double  salt  and  the  two  single  salts  are  in  equili- 
brium with  vapor.  It  is  the  dissociation  curve  for  the  hydrated 
double  salt.  This  curve  must  lie  above  the  dissociation  curves  for 
hydrated  copper  acetate  or  hydrated  calcium  acetate  because  these 
compounds  can  not  begin  to  dissociate  until  the  whole  of  the  double 
salt  has  disappeared.  It  is  to  be  noticed  that  OO,  and  OO3  can  be 
looked  upon  as  dissociation  curves  for  the  hydrated  double  salt 
though  not  in  the  usual  sense  of  the  term.  At  pressures  given  by 
OO,  the  double  salt  effloresces  under  water  with  formation  of  copper 
acetate  ;  at  pressures  given  by  OOt  it  effloresces  under  water  with 
formation  of  calcium  acetate  while  at  pressures  given  by  OF  it  efflo- 
resces with  formation  of  the  two  single  salts.  Only  this  last  is  usu- 
ally looked  upon  as  a  dissociation  curve.  At  pressures  between  OF 
and  OOj  the  hydrated  double  salt  is  stable  in  presence  of  solid  calci- 
um acetate  ;  between  OF  and  OO,  in  presence  of  solid  copper  ace- 
tate. It  was  seen  in  discussing  Fig.  44  that  KE  might  be  looked 
upon  as  a  continuation  of  SK  ;  but  such  a  view  is  untenable  with  a 
hydrated  double  salt  because  it  can  change  under  water  in  two  ways. 
In  this  particular  case  it  is  doubly  impossible  because  OO,  OO,  and 
OF  all  lie  on  the  same  side  of  the  point  O. 

The  temperature  of  the  point  O  is  a  maximum  temperature  for 
this  hydrated  double  salt  whether  in  equilibrium  with  both  solution 
and  vapor  or  not.  The  conditions  under  which  this  occurs  have 
been  formulated  by  Roozeboom*  as  follows  :  "  The  quintuple  point 


van  't  Hoff  and  van  Deventer,  Zeit  phys.  Chem.  I,  173  (1887). 
'Ibid.  I,  227(1887). 
'Ibid,  a,  ^17  (1887). 


1 9o 


The  Phase  Rule 


which  one  meets  in  the  study  of  hydrated  double  salts  is  only  a  tran- 
sition point  (maximum  temperature)  for  the  double  salt  if  the  double 
salt  contains  more  water  of  crystallization  than  the  two  components 
together  and  if  the  change  into  the  components  and  solution  is  ac- 
companied by  contraction  ;  in  all  other  cases  the  hydrated  double 
salt  can  exist  at  higher  and  lower  temperatures  than  the  quintuple 
point." 

As  an  example  of  a  hydrated  double  salt  which  can  exist  in  equi- 
librium with  solution  only  above  a  certain  temperature,  Roozeboom 
has  taken  the  double  sulfate  of  sodium  and  magnesium.1  In  Fig.  47 
is  the  pressure-temperature  diagram  for  the  equilibrium  around  the 
quintuple  point  at  21.5°.  The  monovariant  systems  containing  both 

P 


FIG.  47. 

solution  and  vapor  phases  are  represented  by  OjO  with  the  hydrated 
double  salt,  Na.2Mg(SO4)24H2O,  and  hydrated  sodium  sulfate, 
Na2SO4ioH2O,  as  solid  phases  ;  by  OjO.2  with  double  salt  and  hy- 
drated magnesium  sulfate  ;  by  OjD  with  the  two  hydrated  single 
salts  as  solid  phases.  These  three  curves  end  with  the  appearance 
of  anhydrous  sodium  sulfate,  magnesium  sulfate  with  six  of  water 
and  ice,  respectively.  Along  O^  the  phases  are  the  two  single 
salts,  the  double  salt  and  solution.  This  curve  slants  to  the  right  be- 
cause the  change  of  the  single  salts  into  double  salt  and  solution  is 
accompanied  by  an  expansion  of  volume.2  The  curve  O1F1  is  the 

1  Zeit.  phys.  Chein.  2,  513  (1888). 

7  van  't  Hoffand  van  Deventer,  Ibid.  I,  173  (1887). 


Three  Components  191 

pressure-temperature  curve  for  the  three  solid  phases  and  vapor.  It 
is  clear  from  the  diagram  that  the  point  O,  is  not  a  minimum  tem- 
perature for  the  double  sulfate  of  sodium  and  magnesium.  A  quin- 
tuple point  may  be  a  maximum  or  a  minimum  temperature  for  a 
given  solid  phase  in  equilibrium  with  solution  and  vapor,  as  has 
already  been  shown  in  the  chapter  on  quintuple  points  ;  but  if  it  is 
not  required  that  the  solid  phase  shall  coexist  with  solution  and  va- 
por, the  quintuple  point  is  not  a  minimum  temperature  for  any  solid 
phase,  while  it  is  a  maximum  temperature  for  a  hydrated  double  salt 
only  under  the  conditions  laid  down  by  Roozeboom,  and  the  addi- 
tional one — also  stated  explicitly  by  him — that  the  double  salt  is  not 
stable  at  its  melting  point. 

Having  studied  the  arrangement  of  the  curves  round  a  quintu- 
ple point  we  are  in  a  position  to  consider  the  pressure-temperature 
diagram,  including  all  the  monovariant  systems  in  which  a  hydrated 
double  salt  is  one  of  the  solid  phases.  It  has  already  been  shown 
that  the  field  for  a  ternary  compound  must  be  bounded  by  three 
curves,  and  may  be  by  many  more.  The  fields  for  the  salts 
K2Mg(SO4)26H2O  and  CuCl22KCl2H2O  were  each  bounded  by  four 
curves  ;  there  are  no  instances  known  as  yet  in  which  the  field  is 
bounded  by  three  curves,  but  such  a  state  of  things  is  readily  possi- 
ble and  must  be  taken  into  account  because  it  is  the  simplest  case. 
It  may  be  \vell  to  mention  that  two  salts  which  always  crystallize  in 
the  anhydrous  form  from  aqueous  solution  can  not  form  a  hydrated 
double  salt  unless  this  compound  can  exist  in  stable  equilibrium  with 
ice,  solution  and  vapor.  This  can  easily  be  seen  by  constructing  the^ 
appropriate  diagram. 

In  Fig.  48  are  the  pressure-temperature  curves  for  a  hydrated 
double  salt  which  has  no  true  melting  point  and  which  can  exist  in 
stable  equilibrium  with  only  three  different  solid  phases,  one  of  these 
being  a  hydrated  salt.  The  figure  is  merely  a  combination  of  the 
last  four  diagrams.  In  order  to  make  the  relation  between  this  and 
the  concentration-temperature  diagram  more  clear,  the  latter  is  shown 
with  the  same  lettering  in  Fig.  49.  If  we  call  water  A,  one  of  the 
salts  B  and  the  other  C,  the  hydrated  salt  will  be  AB  and  the  hy- 
drated double  salt  ABC.1  In  Fig.  48  the  curve  OD  is  the.pressure- 

1  In  the  salts  AB  aud  ABC  the  letters  denote  the  qualitative  and  not  the 
quantitative  relations. 


IQ2 


The  Phase  Rule 


temperature  curve  for  B,  C,  solution  and  vapor  ;  OO1  for  ABC,  C, 
solution  and  vapor  j1  OO2  for  ABC,  B,  solution  and  vapor  ;  OjO2  for 
ABC,  AB,  solution  and  vapor  ;  OjDj  for  AB,  C,  solution  and  vapor  ; 
OE  for  ABC,  B,  C  and  solution  ;  C^E,  for  ABC,  AB,  C  and  solu- 
tion ;  OF  for  ABC,  B,  C  and  vapor  ;  C^F,  for  ABC,  AB,  C  and  va- 
por. These  curves  have  been  considered  under  the  systems,  water 
with  sodium  and  magnesium  sulfates,  and  water  with  copper  and 
calcium  acetates.  O2  is  also  a  quintuple  point  analogous  to  the  one 
in  Fig.  44.  There  are  therefore  the  curves  O2D2  for  AB,  B,  solution 
and  vapor  ;  O2E2  for  ABC,  AB,  B  and  solution  ;  O2F2  for  ABC,  AB, 
B  and  vapor.  The  relations  of  the  three  curves  OF,  O,F2  and  OjFj 


FIG.  48. 

and  the  changes  in  the  monovariant  systems  when  the  pressures  fall 
below  those  of  the  curves  are  well  worth  a  brief  discussion.  Whether 
«any  two  of  the  three  curves  can  intersect  is  not  known.  If  they  do 
not,  OjF,  will  always  lie  above  O2F2  and  this  latter  always  above  OF. 
Along  OF  the  solid  phases,  as  we  have  already  seen,  are  ABC,  B  and 
C.  If  the  pressure  falls  below  the  limiting  value,  the  hydrated 
double  salt  will  effloresce  with  formation  of  the  salts  B  and  C.  Along 
O2F2  the  solid  phases  are  ABC,  AB,  and  B.  Decrease  of  pressure 
will  cause  the  hydrated  salt  AB  to  effloresce  with  formation  of  the 
divariant  system,  ABC,  B  and  vapor.  This  system  will  remain  un- 
changed until  the  pressure  reaches  the  value  for  OF  at  that  tempera- 


This  line  has  accidentally  been  omitted  from  the  diagram. 


Three  Components  ,93 

ture,  when  the  hydrated  doable  salt  will  begin  to  change  into  B  and 
C.  Along  O,F,  the  solid  phases  are  ABC,  AB  and  C.  Here  the  hy- 
drated doable  salt  will  effloresce  ;  but  not  with  formation  of  B,  as 
might  be  expected.  It  must  be  remembered  that  the  salt  AB  con- 
tains more  water  of  crystallization  than  the  ternary  compound  ABC.1 
When  AB  effloresces  in  presence  of  C,  there  is  formed  the  hydrated 
double  salt  ABC.  If  there  is  an  excess  of  AB  relatively  to  C  in  the 
monovariant  system,  ABC,  AB,  C  and  vapor,  a  continued  decrease 
of  the  external  pressure  will  cause  formation  of  the  divariant  system 


FIG.  49. 

ABC,  AB  and  vapor  ;  then  of  the  monovariant  system  ABC,  AB,  B 
and  vapor  ;  the  salt  AB  will  next  disappear  :  then  the  salt  C  will  ap- 
pear, and  eventually  ABC  will  disappear  leaving  only  B,  C  and  the 
vapor  of  A.  If  there  is  an  excess  of  C  relatively  to  AB  in  the  mono- 
variant  system,  ABC,  AB,  C  and  vapor,  there  will  be  formed  the  diva- 
riant  system,  ABC,  C  and  vapor,  which  will  next  change  into  the 
system  ABC,  B,  C  and  vapor,  with  final  disappearance  of  ABC.  In 
other  words,  with  excess  of  AB  the  system  passes  from  O,F,  through 
O,F,  and  OF  ;  with  excess  of  C  the  system  passes  from  O,F,  through 
OF  without  intermediate  formation  of  the  monovariant  system  exist- 
ing along  OjF,.  The  salt  AB  effloresces  in  different  wa>-s  and  at  dif- 
ferent pressures,  depending  on  its  being  in  the  presence  of  the  salt 
B  or  the  salt  C. 

'See  page  195. 
13 


1 94  The  Phase  Rule 

If  the  field  for  the  ternary  compound  is  bounded  by  four  curves 
instead  of  three,  this  will  make  the  difference  in  the  pressure- 
temperature  diagram  that  one  of  the  lines  OO,,  OO2  and  O,  O2  will 
have  a  break  and  there  will  be  a  new  quintuple  point  at  this  break. 
It  must  be  distinctly  remembered  that  the  typical  diagram  given  in 
Fig.  48  does  not  apply  to  the  case  where  the  hydrated  double  salt  has 
a  stable  melting  point.  If  any  of  the  monovariant  systems  have  a 
temperature  maximum  somewhere  in  the  middle  of  the  curve,  repre- 
senting them,  it  is  clear  that  this  will  alter  the  diagram  very  consid- 
erably. Since  no  case  is  yet  known  of  a  ternary  compound,  made 
from  two  solids  and  a  liquid  which  has  a  true  melting  point  it  is  not 
worth  while  to  consider  the  diagram  for  this  special  case. 

The  relations  between  pressure  and  temperature  have  been  de- 
termined quantitatively  for  only  one  S5rstem.  Vriens1  has  studied 
the  equilibrium  between  cupric  chloride,  potassium  chloride  and 
water  from  this  point  of  view.  One  of  the  interesting  things  about 
his  results  is  that  the  curve  for  CuCl22KCl2H2O,  CuCl22H,O, 
CuCl2KCl  and  vapor  lies  above  the  curve  for  CuCl22KCl2H2O, 
KC1,  CuCl2KCl  and  vapor  while  this  latter  probably  lies  above  the 
undetermined  curve  for  CuCl,2HaO,  CuCl2  CuCl2KCl  and  vapor. 
This  seems  like  a  contradiction  of  the  theoretical  results  already  ob- 
tained ;  but  the  difference  is  due  to  the  formation  of  an  anhydrous 
double  salt  CuCl2KCl.  Starting  from  the  system  CuCl,2KCl2H2O, 
CuCl22H2O,  CtiCLKCl  and  vapor,  and  decreasing  the  external 
pressure,  there  will  be  disappearance  of  copper  dipotassium  chloride 
and  hydrated  copper  chloride  with  formation  of  copper  potassium 
chloride.  If  the  hydrated  double  salt  be  present  in  excess,  hydrated 
copper  chloride  will  be  the  first  phase  to  disappear,  forming  the  di- 
variant  system,  CuCl32KCl2H2O,  CuCl2KCl  and  vapor.  At  a  yet 
lower  pressure  the  hydrated  double  salt  will  effloresce  writh  formation 
of  potassium  chloride  and  copper  potassium  chloride,  the  pressure 
remaining  Constant  so  long  as  the  monovariant  system,  CuCl22KCl 
2H2O,  KC1,  CuCl,KCl  and  vapor  is  present.  If  hydrated  copper 
chloride  were  originally  in  excess,  instead  of  the  hydrated  double 
salt,  this  latter  would  be  the  first  to  disappear,  leaving  the  divariant 


'Zeit.  phys.  Chem.  7,  194  (1891). 


Three  Components  195 

system,  CuCU2H2O,  CuCl,KCl  and  vapor.  This  will  remain  in 
stable  equilibrium  until  the  pressure  falls  below  the  value  for  the 
system  CuCl^H^O,  CuCltKCl  and  vapor.  If  we  start  with  the 
hydrated  double  salt,  alone  or  in  presence  of  either  or  both  of  the 
salts,  KC1  and  CuCl,KCl,  the  hydrated  double  salt  will  effloresce 
with  formation  of  potassium  chloride  and  copper  potassium  chloride. 
It  will  be  noticed  that  the  reactions,  and  the  pressures  at  which  they 
take  place,  are  functions  of  the  nature  and  relative  amounts  of  the 
solid  phases  originally  present.  In  this  particular  system  there  is 
always  disappearance  of  the  hydrated  double  salt.  This  is  not 
necessarily  the  case.  While  there  are  no  experimental  data  as  yet, 
it  seems  fairly  certain  that  a  mixture  of  magnesium  sulfate  hepta- 
hydrate  and  sodium  sulfate  decahydrate  will  effloresce  with  forma- 
tion of  the  hydrated  double  salt,  Xa^IgCSOJ^HjO.  If  the  sodium 
salt  is  present  in  excess,  this  will  then  effloresce  forming  the  an- 
hydrous salt,  and  not  till  this  change  is  completed  will  the  hydrated 
double  salt  begin  to  dissociate.  If  there  is  an  excess  of  magnesium 
sulfate  heptahydrate,  this  salt  will  effloresce  with  formation  of  the 
hexahydrate  and  the  latter  will  probably  effloresce  before  the  double 
salt  does.  In  spite  of  the  diametrically  opposite  behavior  of  these 
two  hydrated  double  salts,  the  same  rule  applies  to  both  cases  and  to 
all  others  where  only  one  of  the  components  is  measurabl)-  volatile. 
Two  solid  phases  containing  three  components  will  effloresce  with 
formation  of  that  solid  phase  which  can  exist  in  equilibrium  with 
them  at  the  next  higher  quintuple  point. 

When  an  anhydrous  and  a  hydrated  salt  combine  to  form  a 
hydrated  double  salt  with  addition  or  subtraction  of  water,  there 
seems  at  first  no  reason  why  there  should  not  be  a  quintuple  point  at 
which  these  three  solid  phases  could  be  in  equilibrium  with  solution 
and  vapor,  yet  this  is  not  possible.  To  take  a  concrete  case  let  us 
assume  that  lead  and  potassium  iodides  form  no  anhydrous  double 
salt,  and  only  one  hydrated  double  salt,  Pbl^KIsH^O.  If  the 
non variant  system,  hydrated  lead  iodide,  potassium  iodide,  lead 
potassium  iodide,  solution  and  vapor,  can  exist  it  will  be  pos- 
sible to  have  these  three  salts  in  equilibrium  with  vapor  over 
a  series  of  temperatures.  A  moment's  consideration  will  show 
that  there  is  no  way  in  which  this  system  can  effloresce  without 


196  The  Phase  Rule 

forming  a  new  solid  phase  and  thus  a  nonvariant  system  capable  of 
existing  over  an  indefinite  range  of  temperature.  Since  this  is  im- 
possible, it  follows  that  a  quintuple  point  with  these  three  salts  as 
solid  phases  is  impossible  and  that  another  solid  phase  must  appear 
before  this  point  is  reached.  Under  the  conditions  assumed  to  exist, 
the  new  phase  would  be  lead  iodide  either  anhydrous  or  with  one  of 
water.  As  a  matter  of  fact,  it  is  probable  that  lead  and  potassium 
iodides  form  a  second  hydrated  double  salt  and  it  is  this  phase  which 
appears.1  The  hydrated  double  chloride  of  copper  and  potassium 
can  be  made  from  potassium  chloride  and  hydrated  cupric  chloride 
without  addition  or  subtraction  of  water.  Here  it  is  known  that  the 
anhydrous  double  salt  CuCl2KCl  appears  as  solid  phase  and  that 
copper  dipotassium  chloride,  potassium  chloride  and  hydrated  cupric 
chloride  cannot  co-exist  in  equilibrium  with  solution  and  vapor. 

The  efflorescence  of  copper  dipotassium  chloride  has  already 
been  discussed  ;  but  the  effect  of  diminished  pressure  on  the  double 
iodide  of  lead  and  potassium  needs  some  consideration.  If  the  solid 
phases  at  one  of  the  quintuple  points  are  lead  potassium  iodide, 
hydrated  and  anhydrous  lead  iodide  and,  at  the  other,  lead  potassium 
iodide,  potassium  iodide  and  anhydrous  lead  iodide,  a  mixture  of 
hydrated  double  salt  and  hydrated  lead  iodide  will  effloresce  to 
hydrated  double  salt,  hydrated  and  anhydrous  lead  iodide.  When 
the  hydrated  lead  iodide  has  entirely  disappeared  the  hydrated  double 
salt  will  begin  to  effloresce,  forming  anhydrous  lead  iodide  and  po- 
tassium iodide.  The  pure  hydrated  double  salt  or  a  mixture  of  this 
with  potassium  iodide  will  effloresce  with  the  formation  of  the  mono- 
variant  system,  hydrated  double  salt,  anhydrous  lead  iodide  and  po- 
tassium iodide,  the  double  salt  finally  disappearing  entirely. 

If  the  solid  phases  at  one  of  the  quintuple  points  are  lead  potas- 
sium iodide,  a  second  double  salt  with  or  without  water  of  crystalliza- 
tion and  hydrated  lead  iodide  and,  at  the  other,  the  two  double  salts 
and  potassium  iodide,  a  mixture  of  lead  potassium  iodide  and 
hydrated  lead  iodide  will  effloresce  with  formation  of  the  second 
double  salt  at  the  expense  of  the  first,  assuming  that  the  ratio  of 
lead  to  potassium  is  the  same  in  the  two  double  salts.  When  lead 

1  Schreinemakers,  Zeit.  phys.  Cheni.  ro,  471,  (1892). 


TTiree  Components  197 

potassium  iodide  has  entirely  disappeared,  the  hydrated  lead  iodide 
will  begin  to  effloresce.  If  the  second  double  salt  contain  water  of 
crystallization  and  an  anhydrous  lead  potassium  iodide  cannot  exist, 
the  second  double  salt  will  begin  to  effloresce  when  all  the  hydrated 
lead  iodide  is  gone,  and  there  will  be  formed  the  divariant  system, 
anhydrous  lead  iodide,  potassium  iodide  and  water  vapor.  A  mixt- 
ure of  lead  potassium  iodide,  PbIIKl2HtO,  and  potassium  iodide  will 
effloresce,  on  the  same  assumptions,  with  formation  of  the  second 
double  salt  at  the  expense  of  the  first,  the  second  efflorescing  in  its 
turn,  and  the  final  result  being  the  divariant  system,  anhydrous  lead 
iodide,  potassium  iodide  and  water  vapor.  The  first  double  salt,  if 
pure,  will  form  the  second  salt*  and  then  this  will  change  into  the 
two  single  iodides.  If  the  first  double  salt  contain  less  potassium 
iodide  than  the  second,  it  will  change  to  the  latter  and  lead  iodide. 
It  is  evident  that  an  examination  of  the  products  of  efflorescence  will 
give  definite  information  on  the  nature  of  the  phases  existing  at 
quintuple  points  which  can  not  be  easily  investigated  in  the  usual 
manner. 

It  must  be  remembered  that  all  this  reasoning  is  based  on  the 
assumption  that  the  dissociation  curves  do  not  intersect.  Since  two 
adjacent  dissociation  curves  always  have  two  solid  phases  in  common, 
an  intersection  would  form  a  new  quintuple  point  at  which  four  solid 
phases  would  be  in  equilibrium  with  vapor.  If  the  four  solid  phases 
be  denoted  by  the  letters  ar,  jr.  y  and  2  respectively,  the  five  curves 
meeting  in  the  quintuple  point  will  represent  the  simultaneous  pres- 
sures and  temperatures  for  the  five  monovariant  systems ;  a%  x,  y 
and  vapor ;  x,  y,  z  and  vapor ;  y,  z,  w  and  vapor ;  z,w.x  and  vapor ; 
•w,  x,  y  and  z.  No  instance  of  such  a  quintuple  point  has  yet  been 
observed.  If  such  an  one  shall  ever  be  found,  there  will  then  be  a 
temperature  below  which  a  given  compound  can  not  exist.  Such 
cases  occur  in  one  component  sj'stems,  where  one  solid  modification 
changes  into  another  with  evolution  of  heat  and  concentration  of 
volume.  In  binary  systems  no  such  case  has  yet  been  found,  so  that 
it  is  not  surprising  that  none  is  known  for  ternary  systems. 


1  Assuming  that  the  ratio  of  lead  to  potassium  is  the  same  in  the  two  doable 
salts. 


198  The  Phase  Rule 

Before  leaving  the  subject  of  pressures,  reference  should  be  made 
to  one  other  pressure-temperature  curve  for  three  solid  phases  and 
vapor.  By  heating  together  lead  oxide  and  ammonium  chloride,  there 
are  formed,  in  addition  to  the  two  original  substances,  lead  oxide  hy- 
drochloride,  PbOHCl,  and  ammonia  gas.  The  components  are  three 
in  number  :  lead  oxide,  ammonia  and  hydrochloric  acid.  It  was  found 
by  Isambert1  that  the  three  solid  phases  and  vapor  can  exist  over  a 
range  of  temperatures  but  under  only  one  pressure  for  each  temper- 
ature. 


Comptes  rendus,  IO2,  1313  (1886). 


CHAPTER   XIV 

•  SOLID  SOITTIOXS 

Xo  quintuple  points  hare  been  determined  for  systems  in  which 
solid  solutions  are  possible.  This  is  not  because  such  points  are 
rare,  but  because  no  one  has  been  interested  in  them.  While  all  sys- 
tems containing  two  salts  and  water  can  form  one  or  more  non  variant 
systems  under  suitable  conditions  of  temperature  and  pressure  when 
no  solid  solutions  are  possible,  this  is  not  necessarily  true  if  this 
restriction  be  dropped.  If  two  substances  form  a  single  continuous 
series  of  solid  solutions,  it  has  already  been  seen  that  a  quadruple 
point  is  impossible.  Such  a  system  plus  water  could  form  a  quintu- 
ple point  only  in  case  the  water  decomposed  the  solid  solutions. 

Though  no  ternary  systems  containing  solid  solutions  have  been 
studied  in  detail,  there  are  some  rather  haphazard  measurements  by 
Le  Chatelier1  upon  mixtures  of  melted  salts,  while  isotherms  have 
been  determined  bjr  Roozeboom,*  Stortenbeker.3  and  others.*  More 
interesting  from  the  quantitative  than  the  qualitative  point  of  view 
are  the  investigations  of  Knster.'  Ether  is  soluble  to  a  certain  ex- 
tent in  water  and  in  rubber,  while  the  latter  substances  are  practically 
non-miscible.  Kiister  studied  the  di variant  system,  vapor,  solution 
of  ether  in  rubber  and  solution  of  ether  in  water.  The  measure- 
ments extend  only  over  a  limited  range  of  concentrations,  and  the 
amount  of  water  which  was  carried  into  the  solid  phase  by  the  ether 
was  not  determined,  though  it  is  safe  to  assume  that  the  amount  was 
less  than  would  be  calculated  from  the  solubility  of  water  in  ether. 
Similar  measurements  have  been  made  with  starch,  iodine  and  water. 


1  Comptes  rondos,  D«,  415  (1894)- 
-*  ZeiL  phjs.  Chern.  8,  53*  (1891)  : 


'Ibid.  16,  230;  17.  6*3  (1895). 

'  Pock,  Ibid.  I*.  661  (1893)  ;  Muthmann  and  Kontze,  Zrit.  Krrst.  *ft  J0 

(1894). 

*Z«t.  phys.  Chrm.  IJ,  445  (1894). 


200  The  Phase  Rule 

showing  that  the  iodine  forms  a  solid  solution  with  starch  and  that 
the  system  is  therefore  divariant.1  Walker  and  Appleyard2  have 
shown  that  picric  acid  forms  a  solid  solution  with  silk,  and  they  have 
studied  the  distribution  of  picric  acid  between  silk  and  water.  Curi- 
ously enough,  after  giving  a  most  satisfactory  proof  that  a  solid  solu- 
tion is  formed,  they  draw  the  conclusion  that  this  is  not  the  case.  In 
most  other  experiments  with  dye-stuffs  there  are  more  than  three 
components,  because  the  bath  is  acidified  in  order  to  get  better 
results.  Other  cases  of  divariant  systems  in  which  one  of  the  phases 
is  a  solid  solution  have  been  studied  by  Schmidt3  and  by  van  Bem- 
melen.4  Charcoal  possesses  the  power  of  taking  coloring  matter  out  of 
solutions  and  it  was  shown  by  Graham5  that  metallic  salts  are  also 
absorbed.  In  both  these  cases  there  can  be  little  doubt  but  that  solid 
solutions  are  formed  though  adsorption  phenomena  may  take  place 
to  a  minor  extent.  Curiously  enough,  almost  no  work  has  been 
done  in  this  field  of  late  years  and  we  are  very  ignorant  of  the  laws 
describing  these  phenomena.  We  do  not  even  know  whether  the 
behavior  of  charcoal  and  filter  paper  is  analogous  to  that  of  glass 
wool6  though  this  is  probably  not  the  case. 


1  Liebig's  Annalen,  283,  360  (1894). 

2  Jour.  Chem.  Soc.  69,  1334  (1896). 
3Zeit.  phys.  Chem.  15,  56  (1894). 

4  Jour,  prakt.  Chem.  23,  324,  379  (1881)  ;  Zeit.  phys.  Chem.  18,  331  (1895). 
5Pogg.  Ann.  19,  139  (1830). 
6  Cf.  Ostwald,  Lehrbuch  I,  1084. 


CHAPTER  XV 

ISOTHERMS 

Having  already  considered  the  changes  of  concentration  with  the 
temperature  for  a  series  of  typical  systems,  we  can  now  take  up  the 
more  special  cases  of  the  changes  of  concentration  when  one  of  the 
salt  components  is  added  continuously  to  a  solution  kept  at  a  constant 
temperature.  If  we  ignore  the  water  of  crystallization,  which  the 
solid  phases  may  contain,  there  are  eight  cases  to  consider1  : 

Case  I.  The  two  salts  form  neither  double  salts  nor  mix  crys- 
tals. Examples  of  this  are  silver  nitrate  and  acetate,  sodium  chlo- 
ride and  nitrate. 

Case  II.  The  salts  form  a  continuous  series  of  solid  solutions. 
An  example  of  this  is  to  be  found  in  ammonium  and  potassium  sul- 
fates,  permanganate  and  perchlorate  of  potassium. 

Case  III.  There  are  formed  two  series  of  solid  solutions.  Ex- 
amples :  Thallium  and  potassium  chlorates,  cobalt  and  copper  sul- 
fates,  iron  and  magnesium  sulfates. 

Case  IV.  The  solid  phases  are  one  double  salt  and  two  single 
salts.  Examples  :  Potassium  and  copper  sulfates,  sodium  and  mag- 
nesium sulfates. 

Case  V.  The  solid  phases  are  two  double  salts  and  two  single 
salts.  Examples  :  Copper  and  potassium  chlorides. 

Case  VI.  The  solid  phases  are  one  salt,  one  double.salt  and  one 
series  of  solid  solutions.  Example  :  Ammonium  and  ferric  chlo- 
rides. 

Case  VII.  The  solid  phases  are  one  double  salt  and  two  series 
of  solid  solutions.  Examples  :  Potassium  and  silver  nitrates,  am- 
monium and  silver  nitrates,  potassium  and  silver  chlorates,  potassi- 
um and  sodium  sulfates. 


1  Cf.  Roozeboom,  ZeiL  phys.  Chem.  8,  519  (1891)  ;  IO,  158  (1892)  ;  Schrei- 
nemakers,  Ibid.  M,  88  (1893)  ;  Stortenbeker,  Ibid.  17,  643  (1895).  For  the 
earlier  literature  on  the  subject,  see  Ostwald,  Lehrbnch  I,  1072-1079. 


202  The  Phase  Rule 

Case  VIII.  There  are  formed  three  series  of  solid  solutions. 
Examples  :  Magnesium  and  copper  sulfates,  zinc  and  copper  sulfates. 

The  typical  diagrams  for  these  eight  cases  are  given  in  Fig.  50. 
The  ordinates  are  reacting  weights  of  one  salt  and  the  abscissae  react- 
ing weights  of  the  other  salt,  both  in  a  constant  amount  of  water.1 


FIG.  50. 

In  most  cases  addition  of  a  salt  to  a  saturated  solution  of  a  second 
salt,  either  the  acid  or  the  basic  radicle  being  common,  produces  a 
decrease  in  the  solubility  of  the  second  salt.  All  the  diagrams  ex- 
cept the  one  for  Case  VI  are  drawn  to  show  this  ;  but  it  should  be 
kept  in  mind  that  this  is  not  necessarily  true.  Exceptions  are  to  be 
found  with  potassium  and  sodium  nitrates,2  potassium  and  lead  ni- 
trates.3 This  affects  the  direction  of  the  curves  but  not  the  movement 
along  them. 

In  the  diagram  for  Case  I,  the  salt  A  is  solid  phase  along  AO 
and  the  salt  B  along  OB.  Addition  of  B  produces  a  precipitation  of 
A,  and  the  system  passes  along  the  curve  AO,  becoming  poorer  in 
A  and  richer  in  B  until,  at  the  point  O,  the  solution  is  saturated  with 
respect  to  B.  There  is  then  present  the  monovariant  system,  two 
salts,  solution  and  vapor.  Further  addition  of  B  produces  no  change 
in  the  concentration  because  the  solution  is  already  saturated  with 

'Cf.  Schreineniakers,  Zeit.  phys.  Chem.  9,  67  (1892). 

2Nicol.  Phil.  Mag.  (5)  31,  369  (1891). 

3  Le  Blanc  and  Noyes,  Zeit.  phys.  Chem.  6,  385  (1890). 


Three  Components  203 

respect  to  B,  and  therefore  the  added  amount  merely  increases  the 
quantity  of  that  component  as  solid  phase.  In  like  manner  con- 
tinued addition  of  A  to  a  solution  saturated  with  respect  to  B  pro- 
duces a  precipitation  of  B,  the  system  passing  along  BO.  At  O  fur- 
ther addition  of  A  has  no  effect  because  the  solution  is  saturated  with 
respect  to  A.  When  the  two  salts  do  not  form  double  salts  or  solid 
solutions,  neither  can  displace  the  other  completely,  and  the  equilib- 
rium reached  by  adding  B  to  a  solution  of  A  is  the  same  as  that 
reached  b3T  adding  A  to  a  solution  of  B.  Quantitative  determinations 
of  several  isotherms  of  this  class  have  been  made  by  Bodlander1  and 
by  Nicol.2 

In  the  diagram  for  Case  II  the  solid  phase  along  AB  is  a  solid 
solution  with  continuously  varying  concentration.  Addition  of  B 
produces  a  change  in  the  composition  of  the  two  solutions,  solid  and 
liquid,  the  ratio  of  B  to  A  increasing  in  both  as  the  system  passes 
along  the  line  AB.  The  final  result  is  a  pure  solution  of  B  to  within 
any  desired  degree  of  accuracy,  though  an  infinite  quantity  of  B 
must  be  added  to  attain  this.  Addition  of  A  to  a  solution  saturated 
with  respect  to  B  produces  the  reverse  change,  practically  the  whole 
of  B  being  precipitated.  When  the  two  salts  form  a  continuous  series 
of  solid  solutions,  either  can  precipitate  the  other  practically  com- 
pletely from  the  solution.  In  practice  this  result  is  reached  more 
quickly  if  the  crystals  are  removed  as  fast  as  formed.  If  they  are 
left  at  the  bottom  of  the  solution,  one  of  the  salts  will  have  to  be 
added  until  the  total  quantity  of  the  first  is  infinitely  small  in  com- 
parison with  the  amount  of  the  second.  An  isotherm  for  ammonium 
and  potassium  sulfates,  a  pair  of  salts  belonging  to  this  class,  has 
been  determined  by  Fock3  while  Muthmann  and  Kuntze  have  studied 
the  permanganate  and  perchlorate  of  potassium.4 

In  the  diagram  for  Case  III  the  solid  phase  along  AO  is  a  solid 
solution  containing  the  salt  A  in  excess,  and  along  BO  a  solid  solu- 
tion containing  an  excess  of  the  salt  B.  Addition  of  B  to  a  solution 


1  Zeit.  phys.  Chem.  7,  360  (1891). 
-  Phil.  Mag.  (5)  31,  369  (1891). 
3  Zeit.  phys.  Chem.  12,  661  (1893). 
*Zeit.  Kryst.  23,  368  (1894). 


204 


The  Phase  Ride 


saturated  with  respect  to  A  produces  a  change  in  the  solutions,  solid 
and  liquid,  until  the  concentration  denoted  by  the  point  O  is  reached. 
The  solution  is  then  saturated  with  respect  to  both  sets  of  solid  solu- 
tions. Further  addition  of  B  causes  a  decrease  in  the  amount  of  the 
solid  phase  in  which  A  is  solvent  and  an  increase  in  the  quantity  of 
the  other  set  of  crystals,  the  concentrations  of  the  three  solutions, 
two  solid  and  one  liquid,  remaining  unchanged.  When  the  crystals 
containing  an  excess  of  A  are  entirely  consumed,  addition  of  B 
causes  a  change  in  the  concentrations  of  both  the  remaining  solutions, 
the  systems  passing  along  OB.  The  final  result  is,  to  all  intents  and 
purposes,  a  solution  saturated  with  respect  to  B  onl}7.  Exactlv  the 
opposite  changes  take  place  on  adding  A  to  a  saturated  solution  of 
B.  The  system  moves  along  the  curve  BO,  stays  at  O  until  the 
crystals  containing  an  excess  of  B  have  disappeared  and  then  passes 
along  AO,  ending  with  what  is  a  pure  solution  of  A  from  the  stand- 
point of  quantitative  analysis.  When  the  salts  form  two  sets  of  solid 
solutions  either  salt  can  displace  the  other  completely.  An  isotherm 
for  potassium  and  thallium  chlorates  has  been  determined  by  Bakhuis 
Roozeboom1  and  for  other  pairs  of  salts  by  Stortenbeker*.  For  a 
lecture  experiment,  the  sulfates  of  cobalt  and  copper  are  excellent 
because  the  two  sets  of  mix  crystals  have  different  colors. 

In  the  diagram  for  Case  IV  the  salt  A  is  solid  phase  along  AO 
and  the  salt  B  along  BN ;  while,  along  ON,  there  is  the  double  salt. 
Addition  of  B  to  a  saturated  solution  of  A  causes  the  system  to  pass 
along  the  line  AO.  At  O  the  double  salt  appears  as  solid  phase  and 
further  addition  of  B  causes  no  change  in  the  concentration  until  all 
of  A  in  the  solid  phase  has  been  converted  into  double  salt.  When 
this  has  happened  the  system  will  pass  along  the  line  ON  until,  at  N, 
B  appears  as  solid  phase  and  further  addition  of  this  component  pro- 
duces no  change  in  the  concentration.  Addition  of  A  to  a  solution 


^eit.  phys.  Chem.  8,  530  (1891).  The  compositions  of  the  two  solid  solu- 
tions at  the  point  O  are  36.3  and  97.93  reacting  weights  of  potassium  chlorate 
in  one  hundred  reacting  weights  of  the  solid  solutions,  the  temperature  being 
10°. 

1  Ibid.  16,  250,  1895.  It  is  not  yet  certain,  that  there  are  not  two  classes 
under  this  one  heading,  two  series  of  solid  solutions  and  limited  solubilities, 
two  series  of  solid  solutions  with  unlimited  solubilities. 


Three  Components  205 

saturated  in  respect  to  B  causes  the  system  to  pass  along  BX  and 
then  along  XO.  the  concentration  remaining  constant  at  X  until  the 
crystals  of  B  have  been  entirely  changed  into  double  salt.  At  O  the 
salt  A  appears  as  solid  phase  and  no  further  change  of  concentration 
is  possible.  When  the  solid  phases  along  the  isotherm  are  the  two 
single  salts  and  a  double  salt,  neither  salt  can  precipitate  the  other 
completely  from  the  solution  and  the  equilibrium  reached  by  adding 
B  to  a  solution  of  A  is  not  the  same  as  that  reached  by  adding  A  to 
a  solution  of  B.  It  is  immaterial  whether  the  double  salt  is  decom- 
posed by  water  or  not.  Sodium  and  magnesium  sulfates  furnish  an 
excellent  example  of  this  case,  the  double  salt  being  decomposed  by 
water  between  21.5°  and  25°  and  not  decomposed  above  the  latter 
temperature.1  Some  points  on  the  curve  for  potassium  and  copper 
snlfate  have  been  determined  by  Trevor.*  This  is  a  case  where  the 
double  salt  is  not  decomposed  by  water  at  any  temperature  so  far  as 
is  now  known.  As  an  illustration  of  a  double  salt  decomposed  by 
water  at  all  temperatures  we  have  .the  double  iodide  of  lead  and  po- 
tassium studied  by  Schreinemakers.*  Though  these  three  double 
salts  behave  so  differently  towards  water,  the  three  systems  show  the 
same  changes  when  either  salt  component  is  added  continuously. 

In  the  diagram  for  Case  V  the  salt  A  is  solid  phase  along  AO 
and  the  salt  B  along  BX.  while  one  of  the  double  salts  exists  along 
OK  and  the  other  along  XK.  Addition  of  B  to  a  solution  of  A 
causes  the  system  to  pass  from  A  to  X,  the  concentration  remaining 
constant  at  O  until  all  the  crystals  have  been  converted  into  the  first 
double  salt,  and  constant  again  at  K  until  the  crystals  of  the  first 
double  salt  have  been  changed  into  those  of  the  second  double  salt. 
When  the  system  has  reached  the  point  X,  further  addition  of  B  pro- 
duces no  change  beyond  an  increase  in  the  amount  of  B  present  as 
solid  phase.  When  the  solid  phases  along  the  isotherm  are  the  two 
single  salts  and  the  two  double  salts,  neither  salt  can  displace  the 
other  completely,  and  the  equilibrium  reached  by  adding  B  to  a  solu- 
tion of  A  is  not  the  same  as  that  reached  by  adding  A  to  a  solution 


Roozeboom,  ZeiL  phys.  Chem.  a,  518  (1888). 
Ibid  7,46011891} 
'  Ibid.  10,467  (1893). 


206  The  Phase  Ride 

of  B.     An  example  of  this  form  of  isotherm  is  furnished  by  copper 
and  potassium  chloride  between  56°and  92°. r 

Only  one  instance  of  Case  VI  has  yet  been  observed,  the  salts 
being  ammonium  and  ferric  chloride.2  Along  AO  the  solid  phase  is 
FeCl36H2O  ;  along  ON  the  double  salt  2NH4ClFeCl3H2O,  and  along 
NB  a  series  of  solid  solutions.  Addition  of  ammonium  chloride  to  a 
saturated  solution  of  ferric  chloride  produces  an  increase  in  the  solu- 
bility of  the  latter  and  the  system  passes  along  the  curve  AO.  At 
O  the  solution  becomes  saturated  with  respect  to  the  double  salt  and 
further  addition  of  ammonium  chloride  brings  about  no  change  in 
the  concentration  until  all  the  crystals  of  ferric  chloride  have  been 
converted  into  crystals  of  the  double  salt.  The  system  then  passes 
along  the  curve  ON  until,  at  N,  there  is  formed  the  monovariant  sys- 
tem, double  salt,  the  first  term  of  the  series  of  solid  solutions,  solu- 
tion and  vapor.  On  adding  ammonium  chloride,  the  crystals  of  the 
double  salt  disappear  while  the  amount  of  the  solid  solution  increases, 
the  concentration  of  the  liquid  and,  the  solid  solutions  remaining  con- 
stant. When  the  double  salt  has  completely  disappeared  the  concen- 
trations of  the  two  solutions  change  with  addition  of  ammonium 
chloride,  approaching  as  a  limit  a  pure  saturated  solution  of  ammo- 
nium chloride.  The  double  salt  is  decomposed  by  water.  The  solid 
solutions  contain  all  three  components.  Addition  of  ferric  chloride 
to  a  solution  saturated  with  respect  to  ammonium  chloride  produces 
formation  of  solid  solutions  and  then  disappearance  of  solid  solutions 
with  formation  of  double  salt,  the  system  passing  along  the  curves 
BNO.  When  the  point  O  is  reached  further  addition  of  ferric  chlo- 
ride has  no  effect  upon  the  concentration  of  the  solution.  When  the 
order  of  solid  phases  along  the  isotherm  is  single  salt,  double  salt 
and  a  solid  solution,  the  second  salt  can  precipitate  the  first  com- 
pletely but  the  first  can  not  do  this  to  the  second. 

In  the  diagram  for  Case  VII  the  solid  phase  along  AO  is  a  solid 
solution  with  A  as  solvent,  along  BN  a  solid  solution  with  B  as  sol- 
vent while  along  ON  double  salt  crystallizes.  Addition  of  B  to  a 
solution  of  A  causes  a  continuous  change  in  the  concentrations  of 


1  Meyerhoffer,  Zeit.  phys.  Chem.  5,  97  ( 1890). 

2  Roozeboom,  Ibid.  IO,  147  (1892). 


Three  Components  207 

the  solid  and  liquid  phases  until  the  system  reaches  O  and  the  double 
salt  crystallizes.  The  concentration  remains  constant  while  the  last 
term  of  the  series  of  solid  solutions  is  being  converted  into  double 
salt.  When  this  is  finished  the  system  passes  along  OX  and  NB, 
the  concentration  remaining  constant  at  X  while  the  double  salt 
changes  into  the  new  series  of  solid  solutions.  The  system  'finally 
approaches  a  saturated  solution  of  pure  B  as  a  limit.  Adding  A  to  a 
solution  of  pure  B,  the  order  of  events  is  reversed  and  the  system  ap- 
proaches a  pure,  saturated  solution  of  A  as  a  limit.  When  the  order 
of  solid  phases  is  solid  solution,  double  salt  and  solid  solution,  either 
salt  can  precipitate  the  other  completely.  Xo  isotherm  of  this  class 
has  been  studied  because,,  in  the  substances  discovered  so  far,  the 
range  of  the  two  solid  solutions  is  so  limited  that  it  requires  a  micro- 
scopic examination  to  prove  that  it  is  not  the  pure  single  salt  which 
separates.  Retgers  has  shown  that  the  nitrates  of  potassium  and  sil- 
ver, the  nitrates  of  ammonium  and  silver,1  the  chlorates  of  potassium 
and  silver,2  and  the  sulfates  of  sodium  and  potassium9  belong  under 
this  heading. 

In  the  diagram  for  Case  VIII,  solid  solutions  occur  along  AO, 
OX  and  XB.  Adding  B  to  the  saturated  solution  of  A  causes  the 
system  to  pass  along  the  curve  AOXB,  while  an  addition  of  A  to  a 
solution  saturated  with  respect  to  B  causes  the  system  to  traverse  the 
path  BXOA.  In  both  cases  the  final  result  is  a  pure  solution  of  the 
component  added,  to  within  any  desired  degree  of  accuracy  ;  and  in 
both  cases  there  is  a  period  of  constant  concentration  at  O  and  at  X, 
while  one  solid  phase  is  being  replaced  by  the  other.  Examples  of 
this  are  magnesium  and  copper  sulfates,  zinc  and  copper  sulfates. 
Retgers*  has  determined  the  limiting  concentrations  of  the  crystals  at 
the  points  O  and  N,  but  there  are  no  determinations  of  the  concen- 
trations of  the  liquid  solutions  at  these  points.  This  win  soon  be 
remedied  as  Stortenbeker*  has  already  announced  a  paper  on  this 
subject.  Stortenbeker*  points  out  that  three  series  of  solid  solutions 
can  occur  in  at  least  two  wavs.  There  can  be  three  different  kinds 


1  Zeit.  phys.  Chem.  4,  611  (1889). 

1  Ibid.  5,  446  (1890).    *  Ibid.  6,  226  ( 1890). 

'Ibid.  15,  573  (1894)- 

*  Ibid.  17,  650  ( 1895 ).  *  Ibid.  I7;  646 1 1895). 


208  The  Phase  Rule 

of  crystals,  as  in  the  two  instances  just  referred  to.  The  same  result 
can  be  reached  with  only  two  kinds  of  crystals  when  a  continuous 
series  of  solid  solutions  is  cut  into  two  parts  by  a  second  series.  An 
example  of  this  is  probably  to  be  found  in  the  sulfates  of  copper  and 
manganese.  Under  these  circumstances  AO  and  NB  would  be  parts 
of  the  same  continuous  curve.  This  case,  as  well  as  Illb,  are  derived 
from  Case  II  by  addition  of  another  series  of  solid  solutions.  In  Illb 
the  new  series  cuts  off  one  end  of  the  other  curve  while  here  it  is  the 
middle  of  the  curve  which  disappears.  When  there  are  three  series 
of  solid  solutions,  either  salt  can  precipitate  the  other  completely.  It 
is  of  no  importance,  so  far  as  this  point  is  concerned,  whether  the 
three  series  are  all  different  or  whether  the  two  end  ones  have  the 
same  crystalline  form. 

In  the  eight  cases  which  have  been  considered  there  have  been 
instances  where  neither  salt  could  precipitate  the  other,  where  only 
one  could  and  where  either  could.  In  the  instance  where  neither 
salt  could  precipitate,  there  are  examples  where  the  same  equilibrium 
is  reached  no  matter  which  salt  is  added  continuously,  and  examples 
where  a  different  equilibrium  is  reached.  This  may  seem  like  a  state 
of  confusion  ;  but  Roozeboom1  has  pointed  out  that  one  simple  rule 
covers  all  cases.  On  adding  a  component  continuously  a  final  equi- 
librium will  be  reached  only  when  the  solution  is  saturated  with  re- 
spect to  that  component.  A  corollary  of  this  is  that  one  salt  can  pre- 
cipitate the  other  completely  only  when  the  second  dissolves  in  the 
first  to  form  a  solid  solution. 


Zeit.  phys.  Chem.  IO,  161  (1892). 


CHAPTER   XVI 

FRACTIONAL  EVAPORATION 

The  change  in  the  system  when  one  of  the  salt  components  is 
added  continuously  at  constant  temperature  is  independent  of  the 
absolute  or  relative  solubility  of  either  of  the  salts,  but  this  is  no 
longer  the  case  when  water  is  added  to  or  removed  from  the  solution. 
It  will  therefore  be  necessary  to  consider  by  itself  the  changes  in  the 
nature  of  the  solid  phase  when  water  is  withdrawn  from  the  solution 
at  constant  temperature,  the  crystals  being  removed  as  fast  as  formed. 

Under  systems  composed  of  two  liquid  components  we  have  al- 
ready considered  the  subject  of  fractional  distillation.  Here,  the 
problem  is  the  somewhat  similar  one  of  fractional  evaporation  of  the 
mother  liquor  at  constant  temperature.1  The  relation  between  the 
liquid  and  the  solid  phase  will  be  seen  most  clearly  if  a  rectangular 
diagram  be  used,  in  which  the  ordinates  represent  reacting  weights 
of  the  salt  A  in  one  hundred  reacting  weights  of  the  salts  A  and  B 
in  the  solution  while  the  abscissae  denote  reacting  weights  of  the 
same  salt  in  one  hundred  reacting  weights  of  the  two  salts  in  the 
solid  phase.1  The  lower  left  hand  corner  of  the  diagram  represents 
a  saturated  solution  of  B  ;  the  upper  right  hand  corner  a  saturated 
solution  of  A.  The  horizontal  double  lines  show  that  the  two  solid 
phases  which  are  connected  can  coexist  in  equilibrium  with  the  solu- 
tion. These  lines  are  not  necessary  to  the  diagram  and  could  be 
omitted.  The  dotted  diagonal  shows  the  solutions  in  which  the  rela- 
tive proportions  of  the  two  salts  are  the  same  as  in  the  solid  phase. 
It  is  clear  that  if  the  relative  concentration  of  A  is  greater  in  the 
liquid  than  in  the  solid  phase  this  difference  will  increase  as  more  of 
the  solid  phase  is  formed  and  the  solution  will  become  relatively  richer 


1  Cf.  Meyerhoffer,  Die  Phasenregel,  49  ;  Schreinemakers,  Zeit.  phys.  Chem. 
II,  81  ( 1893). 

*Cf.  Roozeboom,  Zeit.  phys.  Chem.  8,521  (1891);  Stortenbeker,  Ibid.  16, 
257  (1895)- 
14 


2io  The  Phase  Rule 

with  respect  to  A.  If,  at  any  moment,  the  point  representing  the 
state  of  the  system  lie  above  the  dotted  diagonal,  the  solution  will 
concentrate  on  evaporation  towards  a  pure  solution  of  A  ;  if  it  lie  be- 
low the  line,  it  will  concentrate  towards  pure  B.  In  Fig.  51  are 
given  most  of  the  possible  diagrams  for  fractional  evaporation  at 
constant  temperature  of  a  solution  containing  two  salts.  The  Roman 
numerals  refer  to  the  cases  which  the  pairs  of  salts  illustrate. 

Case  I.  The  solid  phases  are  the  two  single  salts  either  hy- 
drated  or  not  hydrated.  The  diagram  shows  that  where  B  is  solid 
phase  all  the  points  lie  above  the  diagonal  and  the  solution  concen- 
trates toward  A.  All  the  solutions  in  equilibrium  with  A  as  solid 
phase  lie  below  the  diagonal,  and  these  solutions  therefore  concen- 
trate towards  B.  The  final  result  of  fractional  evaporation  in  this 
case  will  be  simultaneous  separation  of  A  and  B,  and  when  this  be- 
gins to  take  place  the  solution  will  go  dry  without  change  of  concen- 
tration. As  an  example  of  this  take  A  =  Nad,  B=  NaNO3,  or 
vice-versa. 

Case  II.  The  salts  form  a  continuous  series  of  solid  solutions. 
There  are  three  possibilities  : 

a.  The  concentration  of  A  in  the  solution  may  always  be  greater 
than  the  concentration  of  A  in  the  solid  phase.     The  solution  changes 
finally   to   a   solution   containing   only    A.       This    happens    when 
A  =  (NH4),SO4,  B  =  K2S(V 

b.  The  concentration  of  A  in  the  solution  may  be  greater  and  then 
less  than  the  concentration  of  A  in  the  crystals.     The  solution  will 
pass  to  the  point  x  where  the  two  concentrations  are  equal  and  will 
then  go  to  dryness  without  change.     This  happens  when  A  =  KMnO4, 
B  =  KC1O4,  or  vice-versa:1 

c.  The  concentration  of  A  in  the  solution  may  be  less  and  then 
greater  than  the  concentration  in  the  crystals.      Solutions  to  the  left 
of  the  point  x  would  concentrate  to  pure  B  ;  solutions  to  the  right  to 
pure  A.     No  instance  of  this  has  yet  been  studied. 

Case  III.  These  are  two  series  of  solid  solutions.  There  are 
three  possibilities : 


'Fock,  Zeit.  phys.  Chem.  *2,  661  (1893). 
IVIuthmann  and  Kuntze,  Zeit.  Kryst.  2£,  368  (1894). 


Three  Components 


211 


L 


^  *N 


ft  \ 


a  \ 


&  N 


212  The  Phase  Rule 

a.  The  concentration  of  A  in  the  solution  may  be  greater  and  then 
less  than  the  concentration  in  the  crystals.     The  solution  will  pass 
to  the  concentration  at  which  the  two  kinds  of  crystals  precipitate 
simultaneously,  and  will  evaporate  to  dryness  without  further  change. 
This  happens   when  A  =  KC1O3,   B  =  T1C1O3,  or  vice-versa  /  also 
with  the  stable  portions  of  the  system  containing  magnesium  and 
ferrous  sulfates  or  manganese  and  cobalt  chlorides.2 

b.  The  concentration  of  A  in  the  solution  may  be  less  and  then 
greater  than  the  concentration  in  the  crystals.     Solutions  correspond- 
ing to  the  first  part  of  the  curve  would  concentrate  to  pure  B,  those 
corresponding  to  the  second  part  of  the  curve  to  pure  A.     The  labile 
modifications  of  the  system  containing  ferrous  and  magnesium  sul- 
fates illustrate  this  form  of  curve.2 

c.  The  concentration  of  A  is  always  greater  in  the  solution  than  in 
the  crystals.     The  solution  finally  changes  to  pure  A.     This  happens 
when  A  =  CoSO47H2O,    B  =  CuSO45H2O  ;3   also  when  A  =  CuCl, 
2KCl2H2O,  B  =  CuCl22NH4Cl2H2O.4 

Case  IV.  The  solid  phases  are  two  single  salts  and  a  double  salt. 
There  are  two  possibilities  : 

a.  The  double  salt  is  not  decomposed  by  water.     The  concentra- 
tion of  A  in  the  solution  is  greater,  then  less,  then  greater  and  finally 
less  than  its  concentration  in  the  crystals.     Solutions  containing  less 
of  A  than  the  double  salt  will  pass  to  the  solution   from  which  the 
double  salt  and  the  salt  B  crystallize  simultaneously.     Solutions  con- 
taining a  larger  proportion  of  A  than  the  double  salt  will  change  to 
the  solution  from  which  the  double  salt  and  salt  A  crystallize  simul- 
taneously while  a  pure  solution  of  the  double  salt  evaporates  to  dry- 
ness  without  change  of  concentration.     This  happens  with  potassium 
and  copper  sulfates  ; "  also  with  sodium  and  magnesium  sulfates  be- 
tween 25°  and  30°. 6 

b.  The  double  salt  is  decomposed  by  water.      The  concentration 


1  Roozeboom,  Zeit.  phys.  Chem.  8,  532  (1891). 

2  9tortenbeker,  Ibid.  16,  250  (1895). 

:!  Unpublished  determination  by  K.  K.  Bosse. 
*Fock,  Zeit.  phys.  Chem.  X2,  658  (1893). 
5  Trevor,  Ibid.  7,  468  (1891). 
"Roozeboom,  Ibid.  2,  518,  (1888). 


Thrct  Components 


213 


of  A  in  the  solution  is  greater  and  then  less  than  the  concentration 
of  A  in  the  solid  phase.  All  solutions  come  finally  to  the  solution 
from  which  the  double  salt  and  salt  B  crystallize  simultaneously. 
This  occurs  when  A  =  XajSO.ioHjO,  B  =  MgSO4jH,O  at  about 
24°  ;'  A  =  KC1,  B  =  CuCl^H/)  at  about  40°  ;*  A  =  CuCl^H^O. 
B  =  LiCl  ;  A  =  Pbl^ELO,  B  =  KI  at  about  69  V  The  behavior  of 
the  system  containing  lead  and  potassium  iodides  is  so  curious  that 
it  is  worth  considering  for  a  moment  in  detail.  Meyerhoffer*  pointed 
out  that,  at  this  temperature,  the  ratio  of  the  two  salts  happened  to 
be  the  same  for  the  two  monovariant  systems,  with  lead  iodide  and 
double  salt  and  with  potassinm  iodide  and  double  salt  as  solid  phases. 
Schreinemakers5  took  this  up  and  showed  that  it  was  necessary  to 
know  the  form  of  the  isotherm  in  order  to  predict  the  changes  and 
that  in  this  particular  case  the  isotherm  was  not  a  line  of  constant 
relative  concentration  for  the  two  salts  as  Meyerhoffer  had  assumed. 
The  diagram  for  this  isotherm  is  given  in  Fig.  52  where  the  concen- 
tration of  lead  iodide  in  the  solid  and  liquid  phase  is  given.  The 
line  for  the  solutions  in  equilibrium  with  the  double  salt  is  really 
double,  there  being  two  sets  of  these  solutions  in  which  the  ratio  of 

J 


-'  Y 


\ 


FIG. 


1  Meyerhoffer,  Zeit-  phys.  Chem.  5,  103,  18901 

*  Meyerhoffer,  Sitzungsber.  Akad.  Wiss.  Wien,  IOI  Lib,  599  (1892). 
'Schreinemakers,  Zeh.  phys.  Chem.  9,  65  (1892). 

*Ibid.  9,  645,  1892 ;  Die  Phasenregel,  52. 

•  Zeit.  phys.  Chem.  ¥O.  475  (1892)- 


214  The  Phase  Ride 

the  two  salts  is  constant  and  only  the  quantity  of  water  varies. 
Starting  from  the  monovariant  system,  lead  iodide,  double  salt,  solu- 
tion and  vapor,  if  we  evaporate  at  the  constant  temperature  of  about 
69°  we  shall  have  the  lead  iodide  converted  into  double  salt,  the  con- 
centration of  the  solution  remaining  unchanged.  When  the  lead 
iodide  has  entirely  disappeared  the  system  will  pass  along  the  line 
YZ  with  precipitation  of  double  salt.  On  reaching  Z  the  system  will 
pass  back  along  the  other  portion  of  the  line  ZY  and  the  double  salt 
will  redissolve  until  the  amount  of  lead  iodide  in  one  hundred  of  the 
two  salts  is  again  expressed  by  the  point  Y.  The  ratio  of  the  water 
to  the  two  salts  is  much  less  than  before  and  the  solution  is  saturated 
both  with  respect  to  potassium  iodide  and  the  double  salt.  The  solu- 
tion will  now  evaporate  to  dryness  without  change  of  concentration. 
When  the  crystals  are  not  removed,  the  order  of  changes  is  :  pre- 
cipitation of  lead  iodide,  conversion  of  lead  iodide  into  double 
salt,  precipitation  of  the  double  salt,  solution  of  double  salt  and, 
finally,  simultaneous  precipitation  of  double  salt  and  potassium 
iodide.  This  has  all  been  pointed  out  very  clearly  by  Schreine- 
makers  in  the  paper  referred  to  ;  but  the  same  changes  will  not  take 
place  with  fractional  evaporation  at  constant  temperature,  the  crystals 
being  removed  as  fast  as  formed.  Under  these  circumstances,  there 
will  first  be  precipitation  of  lead  iodide  and  then  of  double  salt  as  be- 
fore until  the  system  reaches  the  concentration  represented  by  the 
point  Z.  If  the  crystals  of  the  double  salt  are  removed,  as  by  hypo- 
thesis, the  system  cannot  pass  along  the  line  ZY  because  there  is  no 
double  salt  present  to  dissolve  and  the  ratio  of  lead  iodide  to  potas- 
sium iodide  cannot  be  increased  by  precipitation  of  the  latter  salt  be- 
cause the  solution  is  not  saturated  with  respect  to  it.  The  only 
thing  possible  is  that  the  solution  becomes  an  unsaturated  one  with 
respect  to  everything  and  remains  so  until  at  Zt  potassium  iodide 
separates  and  the  solution  finally  concentrates  to  the  point  where 
double  salt  and  potassium  iodide  crystallize  simultaneously1.  The 
order  of  changes  when  one  starts  from  a  solution  rich  in  lead  iodide 
and  performs  a  fractional  evaporation  at  the  constant  temperature  of 


1  While  there  seems  to  be  no  flaw  in  the  deduction,  the  conclusion  is  so  un- 
£xpected  that  it  should  certainly  be  tested  experimentally. 


Three  Components  215 

69°  will  be  precipitation  of  lead  iodide,  precipitation  of  double  salt, 
formation  of  an  unsaturated  solution,  precipitation  of  potassium 
iodide  and  simultaneous  precipitation  of  lead  iodide  and  double  salt. 
If  the  line  YZ,  had  projected  upwards  instead  of  downwards  it  seems 
probable  that  the  order  would  have  been,  precipitation  of  lead  iodide, 
formation  of  unsaturated  solution  and  then  simultaneous  precipita- 
tion of  potassium  iodide  and  double  salt. 

Returning  from  this  digression  we  can  now  take  up  Case  V  with 
its  five  subheads,  the  solid  phases  being  the  two  single  salts  and  two 
double  salts. 

a.  Neither  of  the  double  salts  is  decomposed  by  water.     Solutions 
containing  less  of  A  than  the  first  double  salt  concentrate  to  the  so- 
lution from  which  the  first  double  salt  and  salt  B  crystallize  simul- 
taneously ;  solutions  containing  more  of  A  than  the  first  double  salt 
and  less  than  the  second,  concentrate  to  the  solution  from  which  the 
two  double  salts  separate  simultaneously  :  solutions  containing  more 
of  A  than  either  double  salt,  concentrate  to  the  second  double  salt 
and  salt  A.  while  the  two  solutions  in  which  the  salts  are  present  in 
the  same  ratio  as  in  the  two  double  salts,  evaporate  to  dryness  with- 
out change  of  concentration.     An  example  of  this  is  to  be  found  with 
cupric  and  tetrethyl  ammonium  chlorides  at  3I0.1 

b.  One  of  the  double  salts  is  decomposed  by  water,  the  second  is 
not  and  the  concentration  of  A  is  less  in  the  first  double  salt  than  in 
the  solutions  in  equilibrium  with  it.     Solutions  containing  less  of  A 
than  the  second  double  salt  concentrate  to  the  solution  in  equilibrium 
with  the  two  double  salts  ;  solutions  containing  relatively  more  of  A 
than  the  second  double  salt  concentrate  to  the  solution  from  which 
the  second  double  salt  and  the  salt  A  separate,   while  the  solution 
containing  the  two  salts  in  the  same  proportion  as  the  second  double 
salt  does  not  change  concentration  with  loss  of  water  at  constant  tem- 
perature.    This  system  is  realized  at  80°  when  A  =  CuCl^H.O  and 
B  =  KC1.1 

f.  One  of  the  double  salts  is  decomposed  by  water,  the  second  is 
not ;  and  the  concentration  of  A  in  the  first  or  decomposable  double 


1  Meverhoffer,  Die  Phasenregd,  56. 

*  Meyerhoffer,  ZeiL  phys.  Chem.  5,  103  ( 1890). 


216  The  Phase  Rule 

salt  is  greater  than  in  the  solution  in  equilibrium  with  it.  The  only 
difference  between  this  and  the  last  subhead  is  that  the  solutions 
containing  less  of  A  than  the  second  double  salt  concentrate  to  the 
solution  in  equilibrium  with  the  first  double  salt  and  salt  B.  No  in- 
stance of  this  has  yet  been  found. 

d.  Both  double  salts  are  decomposed  by  water,  an  excess  of  A  go- 
ing into  solution  in  both  instances.     All  solutions  concentrate  to  the 
second  double  salt  and  salt  A.     This  happens  at  60°  when  A  =  CuCl2 
2H2O,  B  =  KC1,1  and  at  80°  when  A  =  MgSO46H2O  and  B  =  KZS<V 
With  this  latter  salt  pair  the  lines  for  the  two  double  salts  will  appear 
as  one  continuous  line,  since  the  ratio  of  the  salt  components  is  the 
same  in  the  two  double  salts. 

e.  Both  double  salts  are  decomposed  by  water,  the  first  with  pre- 
cipitation of  B  and  the  second  with  precipitation  of  A.     All  solutions 
concentrate  to  the  solution  in  equilibrium  with  the  two  double  salts. 
No  instance  of  this  has  yet  been  studied.     It  is  impossible  that  two 
double  salts  should  be  decomposed  by  water,  the  first  with  precipita- 
tion of  A  and  the  second  with  precipitation  of  B.     This  can  be  seen 
by  trying  to  construct  a  diagram  for  such  a  state  of  things. 

Case  VI.  The  solid  phases  are  one  of  the  salts,  a  double  salt 
and  a  series  of  solid  solutions.  There  are  five  possibilities  if  it  is  as- 
sumed that  the  line  for  the  solutions  in  equilibrium  with  the  solid 
solutions  does  not  cut  the  dotted  diagonal  ;  otherwise  the  number  is 
much  larger. 

a.  The  double  salt  is  not  decomposed  by  water,  and  the  concen- 
tration of  A  in  the  liquid  solutions  is  greater  than  in  the  solid  solu- 
tions which  separate  from  them.     Solutions  containing  less  of  A  than 
the  double  salt  concentrate  to  the  solution  in  equilibrium  with  the 
double  salt  and  salt  B  ;  the  solution  corresponding  in  composition  to 
the  double  salt  evaporates  to  dryness  without  change,  while  solutions 
containing  more  of  A  concentrate  to  pure  A.     No  instances  of  this 
are  known. 

b.  The  double  salt  is  not  decomposed  by  water,  and  the  concentra- 
tion of  A  in  the  liquid  solutions  is  less  than  in  the  solid  solutions  in 


1  Meyerhoffer,  Zeit.  phys.  Chem.  5,  103  (1890). 

2  van  der  Heide,  Ibid.  12,  416  (1893). 


Tli re c  Components  217 

equilibrium  with  them.  This  differs  from  the  last  in  that  solutions 
containing  relatively  more  of  A  than  the  double  salt  concentrate  to 
the  solution  in  equilibrium  with  double  salt  and  solid  solution.  No 
instances  are  known. 

c.  The  double  salt  is  decomposed  by  water  with  precipitation  of  A. 
All  solutions  concentrate  to  the  solution  from  which  double  salt  and 
salt  B  crystallize  simultaneously.     This  happens  when  A  =  NH4C1, 
B  =  FeCl36H2O.' 

d.  The  double  salt  is  decomposed  by  water  with  precipitation  of  B 
and  the  concentration  of  A  is  greater  in  the  solutions  than  in  the  mix 
crystals  which  separate  from  them.     All  solutions  concentrate  to  pure 
A.     No  instances  are  known. 

e.  The  double  salt  is  decomposed  by  water  with  precipitation  of  B, 
while  tne  concentration  of  A  in  the  solutions  is  always  less  than  in 
the  mix  crystals  in  equilibrium  with  them.     All  solutions  concen- 
trate to  double  salt  and  solid  solution.     No  instances  are  known. 

Case  VII.  Two  series  of  solid  solutions  separated  by  a  double 
salt.  Since  the  double  salt  is  not  decomposed  by  water  in  any  of  the 
known  cases  it  will  be  simpler  to  exclude  the  alternative  possibility. 
There  are,  then,  only  three  subdivisions  to  consider. 

a.  The  concentration  of  A  in  the  solutions  is  greater  than  in  the 
solid  solutions  which  separate  from  them.     Solutions  containing  less 
of  A  than  the  double  salt  concentrate  to  double  salt  and  the  solid  so- 
lutions in  which  B  is  solvent ;  all  solutions  containing  more  of  A 
than  the  double  salt  concentrate  to  the  double  salt  and  the  mix 
crystals  with  A  as  solvent ;  the  solution  corresponding  to  the  pure 
double  salt  remains  unchanged.     It  seems  probable  that  all  the  salt 
pairs  giving  two  series  of  mix  crystals  and  a  double  salt  come  under 
this  head. 

b.  The  concentration  of  A  in  the  second  series  of  solid  solutions  is 
greater  than  in  the  corresponding  liquid  phases  while  the  reverse  is 
true  for  the  first  series  of  mix  crystals.     This  differs  from  the  last  in 
that  solutions  containing  more  of  A  than  the  double  salt  concentrate 
to  pure  A. 

c.  The  line  for  the  first  series  of  solid  solutions  lies  below  the 


Roozeboom,  Zeit.  pbys.  Cheni.  IO,  145  (1892). 


2i8  The  Phase  Rule 

dotted  diagonal  and  that  for  the  second  above.  The  first  set  of  solu- 
tions concentrates  to  pure  B,  the  second  to  pure  A  while  the  solution 
corresponding  to  the  double  salt  evaporates  to  dryness  without  change 
of  concentration. 

Case  VIII.  Three  series  of  solid  solutions.  As  not  a  single  system 
coming  under  this  head  has  yet  been  studied,  it  is  not  worth  while  to 
consider  in  detail  the  innumerable  possibilities  which  can  arise  when 
no  limitations  are  made.  If  the  assumption  is  made  that  no  line  can 
cut  the  dotted  diagonal  there  are  but  two  subdivisions. 

a.  All  the  lines  lie  above  the  dotted  diagonal.     All  solutions  con- 
centrate to  pure  A. 

b.  The  first  line  counting  from  the  left  lies  above  the  dotted  di- 
agonal, the  second  line  lies  either  above  or  below  it  and  the  third  line 
lies  below  it.     All  solutions  concentrate  to  the  solution  the  two  points 
for  which  lie  on  opposite  sides  of  the  dotted  diagonal. 

It  may  be  well  to  state  explicitly  at  the  end  of  this  discussion 
the  general  rule  enabling  one  to  predict  from  the  diagrams  the  be- 
havior of  the  different  solutions.  All  solutions  on  the  diagonal  or 
with  two  points  on  either  side  of  it  evaporate  to  dryness  without 
change  of  concentration.  All  other  solutions  move  to  the  right  with 
fractional  evaporation  if  the  points  representing  them  lie  above  the 
diagonal,  to  the  left  if  below. 

If,  instead  of  studying  the  successive  crops  of  crystals  from  a 
given  solution,  one  should  recrystallize  the  solid  phases  discarding 
the  mother  liquor,  the  order  of  change  would  be  reversed.  The 
separation  of  the  rare  earths  by  fractional  crystallization  rests  upon 
the  principles  discussed  in  the  last  two  chapters  though  the  problem 
is  usually  complicated  by  there  being  more  than  three  components. 


CHAPTER  XVII. 

TWO  VOLATILE  COMPONENTS 

Most  of  the  work  on  systems  with  three  components  forming 
only  one  liquid  phase  has  been  confined  to  two  salts  and  water  owing 
to  the  fewer  experimental  difficulties  attending  such  researches.  To 
Bakhuis  Roozeboom  we  are  indebted  for  a  careful  study  of  the 
system  made  up  of  ferric  chloride,  hydrochloric  acid  and  water.1 
One  of  the  components  is  a  gas  at  ordinary  temperatures  and  press- 
ures and  the  system  is  also  interesting  because  there  is  an  unusually 
large  number  of  binary  compounds  possible  and  all  have  true  melting 
points.  The  results  of  the  investigation  are  expressed  graphically  in 
Fig-  53  where  the  reacting  weight  of  ferric  chloride  is  assumed  to 
correspond  to  the  formula  FeClr  This  was  not  done  in  considering 
the  equilibrium  between  ferric  chloride  and  water  ;  but  it  is  thought 
advisable  here  in  order  to  make  the  scale  of  the  diagram  more  satis- 
factory. The  field  for  ice  as  solid  phase  is  bounded  by  AG'A',  for 
ferric  chloride  with  twelve  of  water  by  AG'YVLC  ;  for  ferric  chlor- 
ide with  seven  of  water  b}*  CL,ME  ;  for  ferric  chloride  with  five  of 
water  by  EMNG  ;  for  ferric  chloride  with  four  of  water  by  GXSOJ 
while  anhydrous  ferric  chloride  exists  to  the  right  of  JOZ.  At  the 
points  B,  D,  F  and  H  the  solutions  have  the  same  concentrations  as 
the  hydrates  with  twelve,  seven,  five  and  four  of  water  respectively. 
The  temperatures  of  the  points  along  this  axis  are  A  —  55°,  B  -f  37°, 
C  27.4°,  D  32.5°,  E  30°,  F  56°,  G  55°>  H  73.5°  and  J  66°.  Along 
the  hydrochloric  acid  side  of  the  triangle  we  have  as  solid  phases  the 
trihydrate  bounded  by  A'G'YWH'C';  the  dihydrate  bounded  by 
C'  H'  K'  E'  and  the  monohydrate  existing  in  the  field  above  E'  K'  T  L'. 
At  B'  and  at  D'  the  solution  has  the  same  composition  as  the  hydrate 
which  exists  as  solid  phase1.  The  temperatures  of  the  points  on 

'Roozeboom  and  Schreinemakers,  Arch.  ne"erl.  29,  95  (1894)  ;  Zeit.  phys. 
Chem.  15,  588(1894). 

1  There  must  be  a  misprint  in  Roozeboom's  figures  for  the  concentration  at 
(y,  for  the  trihydrate  of  hydrochloric  acid  has  a  true  fusion  point.  The  tri- 
angular diagram  of  Roozeboom,  Zeit.  phys.  Chem.  15,  626  (1894),  is  in  accord- 
ance with  this  fact. 


220 


The  Phase  Rule 


FIG.  53- 

this  axis  are  A'  —  90°,  B'  —  25°,  C'  —  35° ,  D'  —  17°. 5  and  E'  —  20°. 
The  data  for  the  temperatures  are  taken  from  Pickering's  measure- 
ments1 and  the  curves  AG',  A'G',  G'Y,  YW.  WH',  C'H',  H'K', 
E'K',  K'T'  and  TL/  are  hypothetical  ones  put  in  approximately  to 
show  the  general  relations.  There  have  been  three  ternary  com- 
pounds studied,  the  compositions  of  which  may  be  expressed  by  the 
formulas  Fe2Cl62HCli2H2O,  Fe2Cl62HCl8H2O,  and  Fe2Cl64H2O.  The 
first  exists  as  solid  phase  in  the  field  VYWV  ;  the  second  in  the  field 
VWH'K'TSNMLV  while  the  field  for  the  third  is  bounded  on  one 
side  by  L/TSOZ  and  the  other  limits  have  not  been  determined  ow- 
ing to  the  difficulties  involved  in  experimenting  in  sealed  tubes.  The 
points  W,  Y,  V,  I/,  M,  N,  S  and  O  represent  the  concentrations  of 
non variant  systems  which  have  been  studied  and  G',  Y,  W,  H',  K'  and 
T  the  concentrations  of  nonvariant  systems  which  were  only  guessed 


Ber.  chem.  Ges.  Berliu,  26,  280  (1893). 


Three  Components  221 

at.  The  temperatures  at  which  these  systems  exist  are  V  —  13°, 
L  -  75°.  M  -  73°,  N  -  16*,  S  -  27.5°,  and  O  +  29°  while  the 
other  values  are  probably  G'  —  100°,  Y  —  60°,  W  —  40°,  H'  —  45°, 
K'  —  55°  and  T  —  65°.  The  line  starting  from  the  lower  left  hand 
corner  of  the  triangle  and  running  to  the  middle  of  the  opposite  side 
is  the  locus  of  all  solutions  in  which  FeCl^  and  HC1  are  present  in 
equivalent  quantities.  The  point  R  represents  a  solution  having  the 
same  composition  as  the  first  ternary  compound  ;  Q  is  the  correspond- 
ing point  for  the  second  and  P  for  the  third.  The  point  R  does  not 
He  within  the  field  for  the  compound  Fe2Cllg2HCli2H<O  and  this  sub- 
stance has  therefore  no  stable  melting  point.  Owing  to  the  slowness 
with  which  transformations  take  place  in  systems  containing  ferric 
chloride  Roozeboom  has  been  able  to  determine  the  melting  point  of 
this  compound  which  he  finds  at  — 6°.  The  solution  is  instable  with 
respect  to  ferric  chloride  with  twelve  of  water.  Since  the  line  from 
the  origin  through  Pdoes  not  cut  the  field  for  FejCl^HCliaHX).  this 
salt  will  always  be  decomposed  by  water  with  precipitation  of  ferric 
chloride  with  twelve  of  water.  Matters  are  very  different  with  the 
second  ternary  compound  Fe^Cl^HClSHjO.  The  point  Q  lies  within 
the  field  for  this  salt  as  solid  phase  and  we  have  the  first  instance  of 
a  ternary  compound  stable  at  its  melting  point.  The  temperature  of 
Q  is  —  3°.  The  line  QRP  enters  the  field  for  the  second  ternary 
compound  at  —  10°  and  leaves  it  at  —  26.5°.  Between  these 
temperatures  the  salt  will  not  be  decomposed  by  water  and  only  one 
solution  is  stable  at  each  temperature.  Between  —  3°  and  —  10° 
the  salt  is  not  decomposed  by  water  and  there  are,  at  each  tem- 
perature, two  solutions  with  which  the  compound  can  be  in 
stable  equilibrium.  The  solutions  between  —  3°  and  —  10°  which 
contain  relatively  more  water  than  the  crystals  are  stable  satu- 
rated solutions ;  the  continuous  series  existing  between  —  3°  and 
—  26.5°  are  stable  supersaturated  solutions.  The  line  QRP  enters 
the  field  for  the  third  compound  at  —  26.5°  and  from  that  tempera- 
ture on  this  compound  is  not  decomposed  by  water.  Since  the  right- 
hand  boundaries  for  this  field  are  not  known,  the  conditions  under 
which  the  salt  will  again  be  decomposed  by  water  cannot  be  stated. 
At  the  point  P  the  composition  of  the  solution  is  the  same  as  that  of 
the  crystals.  -  It  is  not  quite  certain  whether  this  point  lies  to  the  left 


222  The  Phase  Rule 

or  the  right  of  the  continuation  of  the  line  OZ.  If  to  the  left,  the 
compound  has  a  stable  melting  point ;  if  to  the  right  an  instable  one. 
The  temperature  of  the  point  P  is  +  45.7°. 

In  this  system  we  get  an  excellent  confirmation  of  the  theorem 
of  van  Rijn  van  Alkemade1  that  the  intersection  of  the  line  connect- 
ing the  melting  points  of  two  solid  phases  with  the  boundary  curve 
for  those  phases  is  a  maximum  temperature  for  that  monovariant 
system.  The  line  from  Q  to  L,  cuts  the  line  L,M  at  a  point  not 
marked  on  the  diagram.  The  temperature  of  this  point  is  —  4.5°  as 
against  —  7.5°  and  —  7.3°  at  L,  and  M.  The  temperature  of  the 
point  at  which  the  line  QP  cuts  ST  is  26.5°  while  S  and  T  represent 
temperatures  of  —  2 7°. 5  and  about  —  65°.  Here  the  branch  OS  is 
very  short  extending  over  a  range  of  only  one  degree.  For  the 
curve  SO  this  branch  is  practically  non-existent  since  the  line  PH 
passes  through  O.  In  the  same  way  the  line  QB  passes  so  close  to 
I,  that  it  is  impossible  to  distinguish  the  two  points.  The  line  QF 
connecting  the  melting  points  of  Fe2Cl62HCl84H2O  and  Fe2Cl65H2O 
does  not  cut  the  stable  part  of  the  boundary  curve  NM  at  'all  and  the 
temperature  maximum  is  therefore  an  instable  one.  It  has  been 
found  to  occur  at  —  5°,  more  than  two  degrees  higher  than  the 
temperature  of  the  point  M  —  7.3°.  The  ternary  compound 
Fe2Cl62HCli2H2O  is  in  a  state  of  instable  equilibrium  at  its  melting 
point  R  and  the  line  QR  does  not  cut  the  field  for  this  compound. 
In  spite  of  this,  Roozeboom  has  succeeded  in  determining  the  tem- 
perature maximum  at  —  10.5°  while  the  highest  temperature  at 
which  the  two  ternary  compounds  can  be  in  stable  equilibrium  is 
—  13°,  at  the  point  V.  At  these  "  fusion  points  for  two  solid 
phases  "  the  isotherms  come  in  contact  externally.2  It  is  possible  to 
have  an  internal  contact  when  at  least  two  of  the  components  are 
common  to  the  two  solid  phases.  This  special  equilibrium  is  realized 
with  Fe2Cl62HCli2H2O  and  Fe2Cl6i2H2O.  The  instable  isotherms 
for  the  ternary  compound  start  from  R  as  a  centre  and  lie  at  first  en- 
tirely inside  the  isotherms  for  the  binary  compound.  The  former 


JZeit.  phys.  Chem.  II,  311  (1893). 

2  From  the  diagram  it  is  clear  that  WH',  H'K'  and  K'T  will  each  show  a 
maximum  temperature  somewhere  on  the  curve.  These  maxima  have  not 
been  determined. 


Three  Components  223 

set  spread  oat  faster  than  the  latter  and  become  tangent  at  the  point 
where  the  prolongation  of  BR  cuts  YV.  The  temperature  at  this 
point  is  — 12.5°  while  that  of  the  point  V  is  —  13°.  The  change  at 
this  point  is  from  the  ternary  compound  into  the  binary  compound 
and  solution  so  that  the  resulting  divariant  system  is  always  binary 
compound,  solution  and  vapor  irrespective  of  the  relative  amounts  of 
the  solid  phases  originally  present.  Such  points  are,  therefore, 
called  "  points  of  transformation  for  two  solid  phases."  '  The  two 
sets  of  temperature  maxima  can  be  distinguished  without  a  knowl- 
edge of  the  isotherms.  In  the  first  case,  the  maximum  lies  between 
the  two  melting  points  and  in  the  second  case  beyond  them.  It  is 
clear  that  points  of  transformation  for  two  solid  phases  cannot  occur 
when  both  solid  phases  have  stable  melting  points. 

The  forms  of  the  isotherms  for  -f  10°  and  —  ioa  have  been 
marked  in  dotted  hues.  The  first  does  not  cut  the  field  for  either  of 
the  first  two  ternary  compounds.  It  passes  through  the  point  R,  the 
temperature  at  which  a  solution  having  this  composition  can  be  in 
equilibrium  with  FCjCl^iaHX)  being  +  10°.  This  same  solution  can 
be  in  instable  equilibrium  with  FejCl^aHCl  121^0  at  3",  as  we  have 
already  seen.  The  isotherm  for  —  10°  cuts  the  field  for  Fe,Cls 
2HC18H,O  passing  round  the  point  Q.  The  isotherm  for  the  interme- 
diate temperature  of  —  4°  would  be  very  like  that  for  +  10®  plus  a 
closed  curve  round  Q.  The  isotherm  for  —  6°  would  have  the  same 
general  form  as  that  for  —  10°  with  the  addition  of  an  instable 
closed  curve  round  R.  The  isotherm  for  30°  makes  a  disconnected 
semi-circular  figure  round  B  and  a  curve  round  D,  meeting  the 
boundary  curve  at  E.  The  isotherm  then  cuts  FX  and  JO.  takes  a 
turn  round  O,  intersecting  OZ,  and  then  passes  up  into  the  field  for 
FejCl^HCLfH^O  and  out  of  the  diagram.  In  all  the  fields  for  the 
binary  compounds  the  ratio  of  hydrochloric  acid  to  ferric  chloride 
passes  through  a  maximum.  A  host  of  other  isotherms  have  been 
determined  by  Roozeboom  and  Schreinemakers  ;  but  they  present  no 
new  features  which  require  discussion  here.  Since  all  the  solid 
phases,  save  one,  are  stable  at  the  melting  points,  it  is  impossible  to 
predict  whether  a  given  quintuple  point  will  be  a  maximum  or  a  min- 
imum temperature  for  any  phase  or  not. 

1  Roozeboom.  Zeit.  phjs.  Chem.  15,  619  (1893). 


224 


The  Phase  Rule 


In  Table  XXIX  are  some  of  the  data  for  the  boundary  curves, 
the  concentrations  being  expressed  as  reacting  weights  of  hydrochlo- 
ric acid,  and  of  ferric  chloride,  FeCl3,  in  one  hundred  reacting  weights 
of  the  solution.  Values  marked  with  an  asterisk  are  estimated  and 
not  determined  directly.  In  Table  XXX  are  the  temperatures  and 
concentrations  for  the  different  nonvariant  systems.  The  values  for 
the  points  G',  Y,  W,  H',  K'  and  T  are  only  approximative. 


TABLE  XXIX 

Temp. 

HC1 

FeCls 

Temp. 

HC1 

FeCls 

Temp. 

HC1 

FeCl3 

Curve  CL 

Curve  JO 

Curve  NM 

4-27.4° 

o.o 

19.6 

4-66.° 

0.0 

36.8 

-  7-3° 

15-2 

18.8 

26.5 

0.2 

19-5 

60. 

6.4 

33-3 

-  5- 

14.4 

18.2 

25- 

1.9 

18.8 

55- 

9.8 

31-5 

Curve  ML 

20. 

4-3 

18.3 

50. 

ii.  6 

30.4 

-   7-3 

15-2 

18.8 

15- 

8.0 

17-5 

40. 

15-2 

28.6 

-  4-5 

14.1 

17-6 

10. 

10.  I 

17.0 

30. 

17.9 

27-3 

-  7-5 

13-5 

16.6 

O. 

12.8 

16.6 

29. 

18.4 

27.1 

Curve  LV 

-  7-5 

13-5 

16.6 

Curve  OZ 

-   7-5 

15-5 

16.6 

Curve   EM 

29. 

18.4 

27.1 

—  10. 

14-5 

14.6 

-(-30.           o.o    23.2 

30. 

18.7 

27.0 

-13- 

16.0 

12.8 

25-          5-5    21.7 

35- 

19.9 

26.4 

Curve  VY 

20.                7.9      20.8 

40. 

21-5 

25-5 

-13- 

16.0 

12.8 

15.             10.4       19.8 

Curve  OS               —12.5 

16.0 

12.0 

IO.             12.2       19.1 

29. 

18.4 

27.1 

-15- 

16.3 

7-4 

—  7.3       15.2     18.8 

25- 

19.5* 

25-4* 

—  20. 

16.1 

5-4 

Curve   GN 

20. 

19.8 

24.2 

—  60.* 

15-5* 

3-0* 

+  55.           o.o    28.9 

10. 

20.O 

22.8 

Curve  VW 

50.           2.3  i  28.6 

0. 

20.  I 

22.1 

-13- 

16.0 

12.8 

44-           7-2 

25-6 

—  10. 

19.8 

21.3 

—  15. 

17-5 

113 

40. 

8.9   24.8 

—  20. 

19-5 

21.  I 

—  20. 

18.4 

6-7 

33.         10.3    24.3 

-27-5 

19.4 

20.6 

—  40. 

19.3* 

4.6* 

30.         ii.  i     23.9 

Curve  SN 

Curve  ST 

25.         12.6    22.8 

—  27-5 

19.4 

20.  6 

-27-5 

19.4 

20.  6 

20. 

13.7      22.0 

—  20. 

18-5 

20.  I 

-26.5 

20.  I 

20.1 

10. 

15-6 

20.8 

—  16. 

17.7 

19.9 

—  29-5 

21.7 

21.9 

o.         16.4 

20.3 

Curve   NM 

-65-5* 

29.9* 

15-5* 

—  10.             17.!'     20.1 

—  16. 

17.7 

19.9 

—  16.         17.7     19.9 

—  10. 

16.6 

19-5 

Three  Components 
TABUJ  XXX 

225 

Temp. 

HC1 

FeClj 

Temp. 

HC1 

FeCl, 

V 

If 

N 
S 
0 

—  7-5 
—  7-3 
—  16. 

—27-5 
+29. 

16.0 
13-5 

:=  - 

19.4 
18.4 

12.8 

16.6 
18.8 
19.9 
20.  6 
27.1 

G' 
Y 
W 
H' 
K' 
T 

-100° 

—  60 
—  40 
—  45 
—  55 
-  65 

12.9 
15-5 
193 
24-5 
29.6 

29-9 

12.4 

4^6 

4-9 
8.6 

15-5 

- 


CHAPTER  XVIIII 

COMPONENTS   AND    CONSTITUENTS 

In  none  of  the  systems  considered  thus  far  has  there  been  any 
difficulty  in  determining  the  number  and  nature  of  the  components  ; 
but  this  is  not  always  the  case.  When  two  compounds  can  react  to 
form  two  others  by  metathesis,  it  is  not  obvious  how  many  compo- 
nents there  are,  while  matters  are  complicated  still  more  by  the  pos- 
sibility of  several  sets  of  components.  This  makes  it  necessar}'  to 
discuss  the  question  of  the  number  and  choice  of  components.1  It 
must  be  kept  in  mind  that  there  are  n  +  2  phases  in  a  nonvariant 
system  only  when  the  n  components  are  independently  variable. 
Gibbs2  starts  from  the  fact  that  a  system  containing  n  independently 
variable  components  and  r  phases  is  capable  of  n  -f-  2  —  r  independ- 
ent variations  when  we  exclude  "passive  resistances  to  change, 
effects  due  to  gravity,  electricity,  distortion  of  the  solid  masses  and 
capillary  tensions.''  From  this  it  follows  that  a  system  with  n  inde- 
pendently variable  components  is  completely  defined  when  there  are 
n  +  2  coexisting  phases.  By  reversing  the  argument  one  can  deduce 
from  the  number  of  possible  coexisting  phases  the  numbers  of  inde- 
pendent variables  ;  but  nothing  about  the  total  number  of  compo- 
nents, for  the  very  simple  reason  that  there  is  no  limit  to  the  num- 
ber which  may  be  selected.  Gibbs8  states  that  there  may  be  any 
number  of  components,  of  which  certain  ones  are  dependency  vari- 
able and  have  no  influence  on  the  number  of  possible  coexisting 
phases.  It  will  simplify  matters  to  introduce  a  new  term  and  to  call 
the  arbitrarily  chosen  substances,  from  which  a  given  system  can  be 
formed,  the  "  constituents  "  4  of  that  system.  We  can  then  use  the 


1  Cf.  also  Roozeboom,  Zeit.  phys.  Chem.  15,  150  (1894)  ;  Wald,  Ibid.  18, 
337  (1895). 

2  Trans.  Conn.  Acad.  3,  152  (1876). 

3  Ibid.  3,  124(1876). 

4Cf.  Trevor,  Jour.  Phys.  Chem.  I,  22  (1896). 


Three  Components  227 

word  ' '  component ' '  to  denote  the  substances  in  the  system  which 
are  capable  of  independent  variation.  For  a  given  system  there  can 
be  different  sets  of  constituents.  To  take  an  extremely  simple  case, 
the  constituents  of  ammonium  chloride  might  be  ammonium  chlo- 
ride ;  ammonia  and  hydrochloric  acid  ;  the  two  radicles,  ammonium 
and  chlorine ;  or  nitrogen,  hydrogen  and  chlorine,  just  as  seemed 
best  under  the  circumstances.  The  number  of  constituents  may 
equal,  exceed,  or  be  less  than  the  number  of  components.  If  there 
are  N constituents  and  h  relations  among  them,  the  number  of  com- 
ponents n  will  be  N —  h.  Stated  in  words,  each  relation  among  the 
constituents  reduces  the  number  of  components  by  one.  This  way 
of  looking  at  the  problem  simplifies  matters  because  we  can  take  as 
constituents  the  chemical  elements,  or  any  groups  of  elements,  and 
the  difference  between  the  sum  of  these  artificially  chosen  constitu- 
ents and  the  number  of  limiting  conditions  gives  the  number  of  com- 
ponents. 

A  few  illustrations  will  make  this  clear.  In  a  system  containing 
hydrogen  and  oxygen,  we  may  say  that  there  are  two  constituents. 
If  there  are  no  limitations,  both  of  these  are  independently  variable 
and  there  are  two  components.  If  the  ratio  by  weight  of  hydrogen 
to  oxygen  is  one  to  eight  in  every  phase,  we  have  introduced  the 
limitation  : 

2H=O, 

and  there  is  but  one  component,  commonly  called  water.  A  system 
containing  potassium,  nitrogen  and  oxygen,  will  have  only  one  com- 
ponent, potassium  nitrate,  if  the  two  limitations  hold  for  all  the 

phases  : 

K  =  N  =  3  O. 

If  we  take  K,  O  and  H  as  the  three  constituents  of  a  certain  system, 
there  will  be  one  component  if  there  are  the  two  limitations  : 

K  =  O  =  H. 
There  will  be  two  components  if  there  is  the  one  limitation  : 

K  -f  H  =  OH. 

In  this  latter  case  the  only  solid  phases  possible  will  be  ice,  potassium 
hydroxide  and  potassium  hydroxide  with  varying  amounts  of  water 
of  crystallization.  If  metallic  potassium,  potassium  oxide,  hydrogen 


228  The  Phase  Rule 

or  oxygen  is  a  possible  phase  under  the  conditions  of  the  experiment, 
the  limiting  condition  does  not  hold  and  there  are  three  components. 
If  K,  NOS  and  Cl  are  taken  as  constituents  in  a  system  composed  of 
potassium  nitrate  and  potassium  chloride,  there  is  one  limitation, 
namely,  that  for  every  phase  : 

K  =  NO3  +  Cl. 

If  the  system  had  been  supposed  to  be  made  up  of  the  chemical  ele- 
ments, there  would  have  been  four  constituents  instead  of  three  ;  but 
there  would  have  been  two  limiting  conditions  instead  of  one,  so  that, 
from  either  point  of  view,  the  number  of  components  is  two.  If  po- 
tassium nitrate  and  potassium  chloride  had  been  taken  as  the  constit- 
uents there  would  have  been  no  limiting  conditions. 

If  we  say  that  the  system,  potassium  nitrate,  potassium  chloride 
and  potassium  bromide,  contains  four  constituents,  K,  NO3,  Cl  and 
Br,  there  is  one  limiting  condition  : 

K  =  NO3  +  Cl  +  Br. 

In  the  same  way  systems  made  from  potassium  nitrate  and  sodium 
chloride  may  be  said  to  have  four  constituents,  K,  NO3,  Na  and  Cl  ; 
but  there  are  only  three  components  because  of  the  limitation  : 

K  +  Na  =  NO3  +  Cl. 

If  the  two  salts  are  taken  in  equivalent  quantities,  there  are  only  two 
components,  the  extra  limiting  condition  being  : 

K  =  NO3  or  Na  =  Cl. 

Potassium  chloride  and  water  contain  four  elements  but  only  two 
components,  if  we  insist  that  in  all  the  phases  there  are  the  relations  : 

K  =  Cl  and  2H  =  O. 

If  the  salt  is  decomposed  by  water  and  the  water  acts  as  a  monobasic 
acid  there  will  be  only  the  one  limitation  ; 

K  +  H  =  Cl  +  OH. 

If  the  water  can  act  also  as  a  dibasic  acid  there  are  no  limiting  con- 
ditions and  the  number  of  components  is  four.  This  state  of  things 
is  difficult  to  realize  at  any  convenient  temperature  with  potassium 
chloride  and  water,  and  it  will  be  more  satisfactory  to  consider  the 
action  of  water  upon  lead  chloride.  At  temperatures  at  which  there 


Three  Components  229 

is  no  decomposition,  the  number  of  components  is  two,  for  there  are 
two  limiting  conditions  which  may  be  expressed  in  two  ways : 

Pb  =  2C1  and  aH  =  O. 
or,   PbCl  =  Cl  and  H  =  OH. 

If  lead  oxide  hydrochloride.  PbOHCl,  can  be  formed  under  the  con- 
ditions of  the  experiment,  there  will  be  three  components  since  there 
is  only  the  one  relation  : 

PbCl  +  H  =  Cl  +  OH. 

If  lead  oxide  can  also  be  formed  there  will  be  four  components  since 
there  are  then  no  limiting  conditions.  If  lead  oxide  is  a  possible 
phase  and  lead  oxide  hydrochloride  is  not,  there  will  be  three  com- 
ponents, the  single  relation  being : 

Pb  +  2H  =  O  +  2C1. 

This  same  limiting  condition  would  seem  to  hold  throughout  if  we 
we  were  to  double  the  formula  for  lead  oxide  hydrochloride.  and  we 
are  thus  confronted  with  the  difficulty  that  a  system  seems  to  contain 
four  components  if  the  simplest  formula  for  one  of  the  compounds  be 
taken  and  three  components  if  the  formula  be  doubled.  This  would 
imply  that  the  number  of  components  could  be  determined  only  after 
we  knew  the  reacting  weights  of  all  the  constituents  or,  taking  it  the 
other  way  round,  that  the  reacting  weights  could  be  determined  from 
a  knowledge  of  the  variance  of  the  system.  The  fallacy  here  is  due 
to  our  forgetting  that  a  formula  is  merely  a  concise  way  of  stating 
known  facts.  If  we  write  the  doubled  formula  of  lead  oxide  hydro- 
chloride  in  the  form  (PbOHCl),  we  imply  that  the  compound  is  an 
addition  product  of  the — possibly  hypothetical — substance  PbOHCl. 
If  we  write  the  formula  PbO^PbCl,  or  PbOPbO,H,O  we  imply 
that  the  compound  is  an  addition  product  of  lead  hydrate  and  lead 
chloride,  or  a  hydrate  of  the  oxide  and  chloride  of  lead.  If  the  first 
view  is  correct  the  system  will  contain  four  components,  because 
there  is  no  relation  among  the  constituents ;  if  either  of  the  other 
views  is  correct,  there  are  only  three  components  because  we  have 

the  relation: 

Pb  -f  2H  =  O  +  2C1. 

In  the  first  case  the  substance  will  dissociate  with  formation  of  hy- 
drochloric acid  ;  in  the  others  with  formation  of  water.  Experi- 


2  30 


The  Phase  Rule 


mentally,  the  first  reaction  is  the  one  that  occurs.  This  same  point 
was  involved  in  the  discussion  of  the  equilibrium  for  K,  O  and  H  on 
page  227,  though  it^  was  not  gone  into  there.  It  was  shown  that  there 
were  three  components  if  potassium  oxide  was  a  possible  phase.  A 
case  where  water  is  only  a  dibasic  acid  is  to  be  found  with  mercuric 
sulfate  and  water.  The  number  of  components  is  therefore  three, 
and  if  we  take  Hg,  SO4,  H2  and  O  as  constituents,  we  have  one  lim- 
iting condition  : 

Hg  +  2H  =  S04  +  O. 

If  we  take  HgO,  SO3  and  H2O  as  constituents,  there  are  no  limiting 
conditions  and  the  number  of  constituents  equals  the  number  of  com- 
ponents. A  system  made  up  of  copper  sulfate,  potassium  sulfate  and 
water  contains  three  components,  under  ordinary  circumstances. 
The  number  of  components  will  be  reduced  to  two  if  in  each  phase 
there  exists  the  relation  : 

CuSO,  =  K2SO4. 

If  the  temperature  is  raised  until  copper  sulfate  is  attacked  by  water 
there  will  be  four  components.  Under  these  circumstances  the  num- 
ber of  components  becomes  greater  than  the  number  of  constituents 
arbitrarily  selected  and  therefore  h  must  be  negative.  Since  the  pos- 
itive value  of  h  denotes  a  limiting  condition,  a  negative  value  must 
denote  a  condition  of  freedom.  Adding  the  assumption  that  copper 
sulfate  reacts  with  water  is  subtracting  a  limiting  condition  or  adding 
a  degree  of  freedom.  This  can  be  seen  more  clearly  if  we  take  po- 
tassium, copper,  hydrogen,  sulfur  and  oxygen  as  constituents,  five 
in  all.  If  neither  of  the  salts  is  decomposed  by  water  there  are  two 
limitations  : 

2K  +  Cu  =  SO4, 

2H  +  2K  +  Cu  =  SO4  -f  O, 

and  therefore  three  components.  If  the  water  can  decompose  the 
salts  the  first  limitation  drops  out  and  there  are  four  components. 
The  system,  made  from  potassium  nitrate  and  sodium  chloride,  has 
already  been  considered  on  the  assumption  that  the  radicles  are  the 
constituents.  It  is  equally  justifiable  to  take  the  two  salts  as  the  con- 
stituents ;  and  then 

h  =  —  z, 

because  the  two  salts  may  react  with  each  other  giving  an  extra  de- 


Three  Components  231 

gree  erf  freedom.  The  number  of  components  is  therefore  three, 
which  is  the  value  found  before,  as  of  coarse  it  must  be.  While  it  is 
convenient  to  take  the  known  chemical  elements  or  groups  of  these 
elements  as  the  constituents,  this  is  not  necessary,  and  the  Phase 
Rule  does  not  rest  in  any  way  upon  the  permanency  of  the  present 
chemical  elements.  If  copper  should  be  separated  into  two  new  ele- 
ments, for  instance,  these  could  be  taken  as  constituents ;  but,  in  the 
systems  in  which  there  occurs  the  substance  which  we  now  call  cop- 
per, there  would  be  the  limiting  condition  that  the  two  new  elements 
must  be  present  in  the  proportions  in  which  they  formed  copper.  In 
the  same  way,  while  there  may  be  an  increase  in  the  number  of  con- 
stituents in  a  given  phase,  owing  to  what  we  call  electrolytic  dissoci- 
ation, there  is  no  change  in  the  number  of  components. 

The  system  composed  of  potassium  nitrate  and  chloride  presents 
several  characteristics  which  have  not  been  met  before.  It  was  first 
pointed  out  by  Mey erhoffer1  that  it  contained  three  components,  and 
not  two  or  four  as  might  readily  have  been  supposed.  He  also  called 
attention  to  the  fact  that  it  is  impossible  to  select  three  substances 
and  to  say  that  these  and  these  only  are  the  three  components.  It  is 
absolutely  immaterial  whether  one  takes  potassium  chloride,  potassi- 
um nitrate  and  sodium  chloride ;  potassium  chloride,  potassium 
nitrate  and  sodium  nitrate  ;  potassium  chloride,  sodium  chloride  and 
sodium  nitrate ;  or  potassium  nitrate,  sodium  chloride  and  sodium 
nitrate  as  the  components.  It  is  impossible  to  use  the  ordinary  tri- 
angular diagram  to  represent  the  states  of  equilibrium.  If  three  of 
the  salts  occupy  the  three  corners  of  the  triangle,  there  is  no  place 
in  the  diagram  for  the  fourth  salt,  although  the  concentration  of  all 
possible  solutions  can  of  course  be  represented.  Roozeboom*  has 
suggested  the  use  of  a  double  diagram  consisting  of  two  equilateral 
triangles  placed  with  bases  together  so  as  to  form  a  lozenge,  the  four 
salts  occupying  the  four  corners.  The  concentrations  of  the  solutions 
may  then  be  expressed  in  either  triangle,  those  of  the  solid  phases  in 
one  or  the  other  but  not  both  of  the  triangles.  This  leaves  the  ver- 
tical axis  for  the  temperature  as  before.  If  one  does  not  care  about 


1  Die  Phasenregel,  60. 

8Z«t.  phy*.  Chcm.  ^  155  (1894). 


232  The  Phase  Rule 

expressing  the  temperature  but  is  willing  to  use  a  solid  figure,  the 
natural  form  of  the  diagram  would  be  a  tetrahedron. 

No  system  of  this  general  type  has  yet  been  studied  ;  but  since 
three  solid  phases,  solution  and  vapor  constitute  a  nonvariant  system, 
the  monovariant  system  which  is  formed  by  withdrawal  of  heat  will 
be  three  solid  phases  and  vapor.  From  this  it  follows  that  there  is 
one  pair  of  salts  which  can  not  exist  in  stable  equilibrium  with  solu- 
tion and  vapor.  For  this  reason,  in  using  Roozeboom's  diagram,  it 
is  necessary  to  place  the  two  salts  which  can  not  coexist  with  solu- 
tion and  vapor  at  the  opposite  ends  of  the  long  diagonal.  There 
may  be  another  inversion  point  at  some  lower  temperature  at  which 
the  four  salts  are  in  equilibrium  with  each  other  and  with  vapor.  No 
such  case  has  yet  been  observed,  though  Spring1  claims  to  have 
shown  the  existence  of  the  monovariant  system,  sodium  sulfate,  ba- 
rium carbonate,  barium  sulfate  and  sodium  carbonate,  at  pressures 
up  to  six  thousand  atmospheres — a  statement  which  must  be  erron- 
eous. 

Mercuric  sulfate  and  water  react  with  formation  of  mercuric  ox- 
ide and  sulfuric  acid.  If  we  consider  these  four  substances  as  the 
constituents  of  a  system  there  will  be  one  limiting  condition  and 
three  components.  With  most  cases  of  metathesis,  that  is  the  only 
convenient  way  of  treating  the  system  ;  but,  in  this  particular  in- 
stance, it  is  possible  to  take  mercuric  oxide,  sulfur  trioxide  and  water 
as  constituents,  and  there  are  then  no  limiting  conditions.  Hoitsema* 
has  considered  the  system  from  this  point  of  view,  and  it  is  a  most 
excellent  one  for  this  particular  case  ;  but  he  makes  the  mistake  of 
suggesting  that  all  systems  containing  basic  salts  should  be  treated  in 
the  same  way,  which  would  neither  be  general  nor  convenient. 
When  sulfuric  acid  is  added  continuously  to  mixtures  of  mercuric 
oxide  and  water,  the  solid  phases  which  appear  successively  at  25° 
are  HgSO42HgO,  HgO2HgSOt2H2O,  HgSO,H2O  and  HgSO4,  while 
at  50°  the  phase  HgSO4H2O  no  longer  appears. 

In  a  series  of  studies  upon  the  alkaline  tartrates  van  't  Hoff 3  has 
introduced  certain  limitations  which  convert  a  system  really  composed 


Bull.  soc.  chitn.  46,  299  (1886). 
Zeit.  phys.  Chem.  17,  651  (1895). 
Ibid.  I,  173  (1887)  ;  17,  49,  505  (1895)- 


Three  Components  233 

of  four  components  into  one  made  up  of  three.  If  we  take  the  sodi- 
um salt  of  dextrorotary  tartaric  acid  and  the  ammonium  salt  of  laevo- 
rotary  tartaric  acid,  we  shall  have  three  components,  for  the  same 
reason  that  potassium  nitrate  and  sodium  chloride  form  a  three-com- 
ponent system.  Adding  water  we  shall  have  four  components,  and 
it  will  require  six  phases  to  make  a  nonvariant  system.  It  is  found 
experimentally  that,  under  the  conditions  of  the  experiment,  there  is 
an  inversion  point  at  27°  where  the  solid  phases  in  equilibrium  with 
solution  and  vapor  are  the  dextrorotary  sodium  ammonium  tartrate, 
the  laevorotary  sodium  ammonium  tartrate,  and  the  sodium  ammo- 
nium racemate.  There  are  only  five  phases  and  therefore  a  limiting 
condition  must  have  been  introduced.  This  is  the  case,  for  it  is  an 
essential  part  of  the  experiment  that  one  starts  with  sodium  ammo- 
nium racemate,  and  the  limiting  condition  is  therefore  : 

Xa  =  XH4. 

Another  inversion  point  occurs  at  35°,  and  here  the  solid  phases  are 
the  sodium  ammonium  racemate,  sodium  racemate  and  ammonium 
racemate,  the  limiting  condition  being  : 

/-C4H4O«  =  </-C4H4O4. 

The  two  limiting  conditions  are  different  in  the  two  cases  and  it  might 
be  thought  that  it  would  not  be  possible  to  observe  the  two  inversion 
points  with  the  same  original  mixture.  It  is  to  be  noticed  that  both 
these  conditions  are  satisfied  in  the  sodium  ammonium  racemate.  and 
therefore  if  we  add  water  to  this  salt  and  follow  the  changes  with  the 
temperature  we  shall  find  two  inversion  points,  one  at  27°  and  the 
other  at  35°.  Starting  with  any  other  mixture  of  the  sodium  ammo- 
nium tartrates  and  water,  we  should  find  an  inversion  point  at  27° 
but  none  at  35°.  If  we  should  start  with  any  mixture  of  sodium 
racemate,  ammonium  racemate  and  water,  not  containing  the  salts 
in  equivalent  quantities,  there  would  be  an  inversion  point  at  35°  but 
none  at  27°.  By  varying  the  conditions  it  is  thus  possible  to  make 
the  system  behave  as  if  it  contained  three  components  at  all  temper- 
atures, three  components  below  a  certain  temperature  and. four  above 
it,  or  three  components  above  and  four  below  another  definite  tem- 
perature. No  experiments  have  yet  been  made  with  the  system  hav- 
ing four  components  at  all  temperatures.  This  would  be  an  inter- 


234 


The  Phase  Rule 


esting  matter  to  work  upon  because  it  is  very  probable  that  each  in- 
version temperature  would  correspond  to  two  nonvariant  systems, 
owing  to  the  dextrorotary  and  the  laevorotary  salts  having  the  same 
solubility.  There  would,  of  course,  be  four  solid  phases  in  equilib- 
rium with  solution  and  vapor  at  these  inversion  points. 

Since  each  limiting  condition  reduces  the  number  of  components 
by  one,  it  is  clear  that  if  for  any  reason  a  limiting  condition  becomes 
inoperative  the  number  of  components  will  be  increased  by  one. 
Such  a.  state  of  things  occurs  whenever  there  is  a  passive  resistance 
to  change.  Whenever  an  expected  rea'ction  does  not  take  place,  it  is 
said  that  there  is  a  passive  resistance  to  change.  This  is  a  definition 
and  not  an  explanation.  In  some  cases  the  system  is  in  a  state  of 
labile  equilibrium.  In  others  it  appears  to  be  in  a  state  of  stable 
equilibrium  .  but  the  accepted  doctrine  is  that  this  is  an  instance  of 
an  immeasurably  low  reaction  velocity,  and  that  the  system  is  not  in 
equilibrium  at  all.1  If  we  bring  in  the  element  of  time,  it  is  possible 
to  apply  the  Phase  Rule  to  these  cases.  If  a  reaction  does  not  take 
place  to  a  measurable  extent  within  the  time  under  consideration,  it 
has  no  effect  as  a  limiting  condition  for  that  time.  For  instance,  a 
mixture  of  hydrogen,  oxygen  and  water  at  ordinary  temperature  is 
to  be  considered  as  having  three  components  if  the  experiment  does 
not  last  over  a  month  because,  for  that  time,  there  is  no  such  reac- 
tion as  : 

2H2  +  02  =  2H20. 

It  is  quite  possible,  if  the  experiment  lasted  a  thousand  years,  that 
the  system  would  have  to  be  treated  as  containing  two  components, 
though  this  is  not  proved. 

When  the  limiting  condition  takes  the  form  of  a  chemical  reac- 
tion, its  effect  is  to  change  the  number  of  components  ;  but  it  is  pos- 
sible to  introduce  conditions  which  will  affect  the  number  of  pressures. 
This  can  be  done  by  dividing  the  system  into  two  parts  by  a  dia- 
phragm permeable  to  one  of  the  components  and  impermeable  to  the 
others.2  For  the  sake  of  simplicity  we  will  consider  a  system  of  two 
components,  and  if  we  have  in  one  compartment  a  solution  and  in  the 


1  Nernst,  Theor.  Chem.  340. 

2Gibbs,  Trans.  Conn.  4cad.  $,  138  (1876). 


Tkree  Components  235 

other  the  component  which  can  pass  through  the  diaphragm,  a  mo- 
ment's consideration  will  show  that  there  can  not  be  equilibrium 
when  the  pressures  on  the  two  sides  of  the  diaphragm  are  equal. 
The  concentrations  in  the  two  liquid  phases  are  not  equal,  nor  in  the 
two  vapor  phases  which  could  be  in  equilibrium  with  them.  There 
wfll  therefore  be  a  tendency  for  one  component  to  pass  through  the 
diaphragm  diffusing:  from  the  pure  liquid  into  the  solution,  and  this 
diffusion  can  be  checked  only  by  increased  pressure  upon  the  solution 
phase.  Our  conclusion  that  »  +  2  phases  constitute  a  nonvariant 
system  was  based  on  the  assumption  that  the  pressure  and  tempera- 
ture were  uniform  throughout  the  system,  or  that  the  number  of  in- 
dependent variables  equalled  the  *  components  plus  the  temperature 
and  pressure.  If  there  are  two  independently  variable  pressures,  the 
total  number  of  independent  variables  is  m ;  -J-  j,  and  it  will  take  this 
number  of  phases  to  constitute  a  nonvariant  system  while  n  +  2  and 
n  +  i  phases  win  be  necessary  for  monovariant  and  di  variant  sys- 
tems respectively.  Before  considering  such  systems  it  wfll  be  well 
to  ask  whether  a  diaphragm  is  possible  which  shall  be  permeable  to 
one  component  and  impermeable  to  the  other.  Experiment  shows 
that  such  a  thing  can  exist,  and  diaphragms  of  this  sort  are  usually 
called  semipermsable, — a  barbarous  term  which  has  been  universal- 
ly adopted.  Tranbe1  showed  that  when  solutions  of  two  substances 
were  brought  together  carefully  there  was  often  formed  a  coherent 
membrane  at  the  surface  separating  the  two  solutions.,  provided  the 
two  substances  reacted  to  give  a  colloidal  precipitate.  These  mem- 
branes were  always  permeable  to  water ;  but  often  impermeable  to 
dissolved  substances.  One  of  the  best  of  these  membranes  is  made 
of  copper  ferrocyanide,  and  this  seems  to  be  impermeable  to  more 
substances  than  any  other  known.*  An  extended  study  of  the  per- 
meability of  different  membranes  is  to  be  found  in  a  paper  by  Wai- 
den.*  Pfeffer3  succeeded  in  giving  these  membranes  the  strength  to 
withstand  pressure  by  precipitating  them  in  the  walls  of  a  porous 
cell.  This  is  very  difficult  to  do  successfully,  and  Pfeffer  alone  has 
succeeded  in  making  satisfactory  measurements.  Semipermeable 


1  Cf.  Ostwald,  Lehrboch  I.  655, 

*  Zot_  phj*.  Ch«n.  10,  699  (1892)- 

It::  ;     ::- 


236  The  Phase  Rule 

membranes  occur  very  largely  in  nature,  having  been  found  in  plant 
cells,  blood  corpuscles,  bacteria,  and  elsewhere  ;  *  but  these  have  not 
yet  been  used  to  measure  directly  the  pressure  caused  by  the  tendency 
of  the  pure  liquid  to  flow  into  the  solution.  This  pressure  is  com- 
monly called  the  osmotic  pressure.  Pfeffer  measured  the  pressure 
necessary  to  keep  the  solution  in  equilibrium  with  water.  The  sys- 
tem which  he  studied  consisted  of  three  phases,  solution,  solvent  and 
vapor.  For  a  case  of  this  sort  we  have  seen  that  n  +  i  phases  con- 
stitute a  divariant  system  and  that  it  is  necessary  to  fix  two  varia- 
bles before  the  system  is  completely  defined.  Pfeffer  found  that  for 
each  temperature  there  could  be  a  series  of  pressures  and  for  each 
pressure  a  series  of  temperatures,  depending  on  the  concentration  of 
the  solution  phase.  If  the  temperature  and  concentration  be  fixed 
there  is  but  one  osmotic  pressure  at  which  the  system  can  be  in  equi- 
librium. The  experiment  of  changing  the  pressure  upon  the  water 
outside  was  not  tried,  but  there  is  no  doubt  that  as  soon  as  the  vapor 
phase  had  disappeared,  the  osmotic  pressure  would  have  varied  with 
varying  external  pressure,  the  temperature  and  the  concentration  be- 
ing kept  constant.  Very  recently  Raoult2  published  a  note  to  the 
effect  that  vulcanized  rubber  is  permeable  to  ether  and  impermeable 
to  methyl  alcohol,  while  pig's  bladder  is  permeable  to  methyl  alcohol 
and  impermeable  to  ether.  If  we  imagine  methyl  alcohol  and  its 
vapor  separated  from  a  solution  of  methyl  alcohol  aud  ether  by  a  dia- 
phragm of  pig's  bladder,  and  this  same  solution  separated  from  ether 
and  ether  vapor  by  a  diaphragm  of  vulcanized  rubber,  there  will  be 
a  tendency  for  methyl  alcohol  to  flow  into  the  solution  from  one  side 
which  can  be  neutralized  by  a  pressure  P;  upon  the  solution.  There 
will  be  a  tendency  for  ether  to  flow  in  from  the  other  side  which  can 
be  neutralized  by  a  pressure  P2  upon  the  solution.  Since  the  solu- 
tion can  not  be  under  two  different  pressures,  one  or  the  other  of  the 
liquids  will  flow  out  of  the  cell  until  Pt  =  P2 ;  i.  e. ,  until  the  osmotic 
concentrations  of  methyl  alcohol  and  ether  in  the  solution  are  the 
same.  In  other  words,  the  final  equilibrium  is  independent  of  the 


1  Cf.  Zeit.  phys.  Chetn.  2,  415  (1888)  ;  3,  103  (1889)  ;  6,  319  (1890)  ;  7, 
529;  8,685  (1891);  16,261  (1895). 
*  Ibid.  17,  737  (1895). 


Three  Components  237 

initial  concentration  in  the  cell.1  This  conclusion  can  be  reached 
more  satisfactorily  by  an  application  of  the  Phase  Rule.  There  are 
five  phases,  liquid  methyl  alcohol,  vapor  of  methyl  alcohol,  solution, 
liquid  ether  and  vapor  of  ether.  There  are  two  diaphragms  and  con- 
sequently two  extra  pressures,  so  that  //  -f-  ^  phases  constitute  a  non- 
variant  system  and  n  +  3  a  monovariant  one.  The  number  of  com- 
ponents being  two,  we  are  discussing  a  monovariant  system  and  it 
follows  that  for  a  given  temperature  there  can  be  only  one  concen- 
tration in  the  solution  phase  for  which  the  system  is  in  eqnilibrium. 

Returning  to  the  simpler  case  with  only  one  diaphragm  and  as- 
suming that  the  external  pressure  upon  the  solution  is  not  great 
enough  to  prevent  the  formation  of  a  vapor  space,  it  is  clear  that  all 
the  liquid  will  diffuse  in  from  outside  and,  at  equilibrium,  there  will 
be  solution,  vapor  above  the  solution  and  the  vapor  outside  the  cell, 
which  of  course  will  not  be  saturated  vapor.  At  temperatures  below 
the  freezing  point  it  would  be  possible  to  have  ice  and  vapor  in  equi- 
librium with  the  solution. 

There  is  no  theoretical  reason  why  there  should  not  be  systems 
with  two  temperatures  instead  of  two  pressures,  if  we  could  find  dia- 
phragms permeable  to  the  components  and  impermeable  to  heat. 
Since  no  diaphragm  is  known  which  is  impermeable  to  heat,  it  is  not 
worth  while  to  discuss  what  would  happen  in  case  this  hypothetical 
membrane  were  permeable  to  the  components. 


My  attention  was  called  to  this  case  by  Mr.  D.  Mclntosh. 


CHAPTER  XIX 

TWO  LIQUID   PHASES 

No  systems  have  yet  been  studied  in  detail  in  which  there  are 
three  components  and  two  liquid  phases.  If  one  starts  at  the  tem- 
perature  T  with  two  components,  A  and  B,  forming  the  system,  solid 
A,  two  solutions  and  vapor,  and  adds  a  third  component,  C,  there 
will  be  formed  a  monovariant  system  which  can  exist  over  a  range 
of  temperature  limited  only  by  the  appearance  of  a  new  phase  or  the 
disappearance  of  an  old  one.  When  the  third  component  is  a  solid 
at  all  the  temperatures  under  consideration  the  effect  of  adding  it  in 
excess  will  usually  be  to  form  the  n  on  variant  system,  solid  A,  solid 
C,  two  solutions  and  vapor.  If  the  temperature  T  is  very  near  the 
fusion  point  of  pure  B,  it  will  be  possible  by  changing  the  nature  of 
C  to  have  nonvariant  systems  of  this  type  formed  in  which  the  solid 
phases  are  A  and  C,  A  and  B,  or  B  and  C.  This  could  be  realized  with 
phenol,  water  and  a  third  component  or  benzene,  water  and  a  third 
component.  L,et  benzene  be  represented  by  A,  water  by  B,  and  the 
third  component  by  C.  If  C  is  a  substance  soluble  in  water  and  very 
sparingly  soluble  in  benzene,  the  solid  phases  will  be  A  and  C.  If  C 
is  soluble  in  benzene  and  very  .sparingly  soluble  in  water,  the  solid 
phases  will  be  A  and  B,  when  C  is  not  present  in  excess  or  B  and  C 
when  it  is.  It  is  to  be  noticed  that  the  temperatures  at  which  these 
nonvariant  systems  exist  are  not  necessarily  lower  than  those  of  the 
binary  nonvariant  systems  from  which  they  are  derived.  If  the  com- 
ponent C  is  soluble  in  A  and  not  soluble  in  B  the  equilibrium  temper- 
ature will  be  lowered  by  addition  of  C.  If  C  is  soluble  in  B  and  prac- 
tically insoluble  in  A  the  temperature  will  rise,  the  amount  varying 
with  the  nature  of  C  and  the  mutual  solubility  of  A  and  B.  When  the 
third  component  is  a  liquid  at  all  temperatures  under  consideration, 
these  are  four  important  cases.  The  liquids  B  and  C  may  be  conso- 
lute,  while  the  liquids  A  and  C  are  sparingly  miscible.  A  nonvari- 
ant system  with  two  liquid  phases  is  impossible.  An  example  of  this 
would  be  found  when  A  denotes  water,  B  ether  and  C  chloroform. 


Thrte  Components  239 

The  liquids  A  and  C  may  be  consolute  while  the  liquids  B  and  C  are 
sparingly  miscible.  The  nonvariant  system  will  have  A  and  B  as 
solid  phases.  This  can  be  realized  when  A  is  benzene,  B  is  water 
and  C  is  chloroform  or  ether.  The  liquid  C  may  be  miscible  in  all 
proportions  with  the  liquids  A  and  B.  While  there  are  no  experi- 
mental data  upon  which  to  base  an  opinion,  it  seems  very  improbable 
that  a  nonvariant  system  can  be  formed  with  the  phases  A,  B,  two 
solutions  and  vapor,  since  the  two  solutions  approach  each  other  in 
composition  with  increasing  addition  of  C,  and  it  is  to  be  expected 
that  the  two  will  become  identical  in  composition  and  cease  to  exist 
as  separate  phases  before  B  separates  as  solid  phase.  An  example  of 
this  furnished  by  the  system,  naphthalene,  water  and  alcohol.  The 
component  C  may  not  be  miscible  in  all  proportions  with  either  of  the 
liquids  A  and  B.  Under  these  circumstances  it  would  be  possible  to 
have  formed  a  nonvariant  system  made  up  of  three  liquid  phases, 
solid  and  vapor.  This  can  be  realized  with  sulfur,  toluene  and  water, 
though  the  temperature  and  pressure  under  which  this  system  can 
exist  have  not  been  determined. 

The  two  solid  phases  which  are  in  equilibrium  with  the  two  so- 
lutions and  vapor  at  the  quintuple  point  may  be  the  pure  components 
or  compounds  or  solid  solutions.  By  adding  ether  to  a  solution  of 
calcium  chloride  it  would  be  possible  to  determine  several  inversion 
points  at  which  the  solid  phases  should  be  two  compounds.  The  ad- 
dition of  water  to  a  mixture  of  benzene  and  iodine  would  give  a  non- 
variant  system  made  up  of  ice,  a  solid  solution  of  iodine  in  benzene, 
two  liquid  phases  and  vapor,  provided  the  iodine  were  present  in 
small  quantities  only.  In  course  of  time  cases  will  be  found  where 
the  two  solid  phases  are  two  sets  of  solid  solutions,  though  no  instance 
of  this  has  yet  been  studied  qualitatively. 

There  are  no  data  for  nonvariant  or  for  monovariant  systems 
containing  two  liquid  phases.  On  the  other  hand,  divariant  systems 
composed  of  two  liquid  phases  and  vapor  have  received  a  great  deal 
of  attention,  though  the  question  has  not  been  taken  up  with  the 
view  of  applying  the  Phase  Rule.1  For  this  reason  most  of  the  ex- 
periments have  been  made  with  dilute  solutions  only,  and  the  con- 


1  Berthelot  and  Jungfleisch,  Ann.  chim.  phys.  (4)  *6,  396  (1872) ;  Nernst, 
Zeh.  phys  Chem.  8,  no  (18911 ;  Jakowkin.  Ibid.  x8,  585  (1895),  etc.,  tic 


240  The  Phase  Rule 

centration  of  but  one  component  has  been  determined.  These  sys- 
tems show  the  characteristics  of  divariant  systems.  If  the  tempera- 
ture alone  is  fixed,  the  concentrations  in  the  three  phases  can  vary. 
If,  in  addition,  the  composition  of  one  phase  is  fixed,  the  concentra- 
tions of  the  other  phases  are  no  longer  variable.  There  are  a  few 
measurements  giving  the  composition  of  one  of  the  liquid  phases.1 
Although  the  systems  studied  contained  only  two  phases,  liquid  and  va- 
por, yet  it  was  the  limiting  liquid  phase  which  was  investigated,  which 
is,  of  course,  the  one  which  can  exist  in  equilibrium  with  a  second 
liquid  phase.  For  this  reason  the  experiments  referred  to  are  really 
studies  of  divariant  systems.  It  was  found  that  the  curves  for  the 
two  solutions  met  at  an  angle  at  the  point  where  the  two  liquid  layers 
became  identical  in  composition.  This  result  is  interesting  as  a  con- 
firmation of  the  assumption  that  the  same  phenomenon  often  occurs 
when  two  liquids  become  consolute.  In  the  latter  case  it  is  very  dif- 
ficult to  determine  whether  the  apparent  intersection  of  the  lines  is 
real  or  due  to  experimental  error.  Where  two  liquid  phases  contain- 
ing three  components  become  consolute,  there  can  be  no  question 
about  the  facts. 

So  far  it  has  been  assumed  explicitly  that  two  of  the  components 
could  form  two  liquid  layers  at  some  temperature  under  considera- 
tion, and  that  the  third  component  merely  displaced  the  equilibrium 
to  a  certain  extent.  There  are  cases  known  which  do  not  come  un- 
der this  head  ;  where  no  two  of  the  components  form  two  liquid  lay- 
ers at  the  temperature  of  the  experiment,  and  this  particular  equilib- 
rium can  occur  only  when  the  three  components  are  taken  together 
in  the  proper  proportions.  Alcohol  and  water  are  miscible  in  all 
proportions  at  ordinary  temperatures  ;  but  the  addition  of  certain 
salts  will  cause  the  appearance  of  two  liquid  layers,  although  these 
salts  can  not  form  two  liquid  phases  with  water  alone  or  with  alcohol 
alone  at  the  same  temperature.  Some  of  the  salts  which  will  do  this 
are  potassium  and  sodium  hydroxides,  sodium  phosphate,  potassium 
carbonate,  sodium  carbonate,  ammonium  sulfate,  sodium  sulfate,  mag- 
nesium sulfate,  manganese  sulfate,  strontium  chloride  and  many  oth- 
ers. The  same  phenomena  occur  with  water  and  lactones  in  presence 


Pfeiffer,  Ibid.  9,  444  (1892)  ;  Bancroft,  Phys.  Rev.  3,  21  (1895). 


Three  Components  241 

of  alkaline  carbonates,  while  most  salts  have  the  power  of  precipitat- 
ing acetone  from  aqueous  solution  at  some  temperature.  In  all  cases 
yet  known  the  tendency  to  form  two  liquid  layers  when  a  third  com- 
ponent is  added  to  two  consolute  liquids  increases  with  rising  temper- 
ature, solutions  which  are  homogeneous  at  one  temperature  separat- 
ing into  two  liquid  phases  when  heated.  At  yet  higher  temperatures 
it  is  probable  that  the  two  solution  phases  would  again  cease  to  exist 
though  there  are  no  experiments  to  show  it. 

The  system,  ammonium  sulfate,  water  and  alcohol,  has  been 
studied  by  Traube  and  Neuberg1  and  by  Bodlander  ;*  but  the  meas- 
urements cover  too  little  ground  to  permit  of  making  a  complete  dia- 
gram. They  are  sufficient  to  enable  us  to  predict  the  general  form 
of  the  isotherm  at  different  temperatures.  At  low  temperatures  the 
only  divariant  system  possible  is  that  of  salt,  solution  and  vapor,  the 
curve  having  a  break  at  the  point  where  the  solvent  changes  from 
water  to  alcohol.  At  higher  temperatures  these  two  portions  of  the 
curve  are  separated  by  the  curves  for  the  divariant  system,  two  solu- 
tions and  vapor.  If  alcohol  be  added  continuously  to  a  saturated 
aqueous  solution  of  ammonium  sulfate,  the  s)Tstem  does  not  pass 
through  this  last  stage.  There  will  first  be  a  precipitation  of  salt  till 
a  definite  concentration  is  reached.  There  will  then  appear  a  second 
liquid  phase  forming  the  monovariant  system,  salt,  two  solutions  and 
vapor.  With  further  addition  of  alcohol  there  will  be  increased  pre- 
cipitation of  salt  and  a  change  in  the  quantities  though  not  in  the 
concentrations  of  the  two  liquid  phases.  When  one  of  the  liquid 
phases  has  disappeared  completely,  the  concentration  of  the  remain- 
ing solution  phase  will  change  with  addition  of  alcohol. 

In  man)7  instances  where  the  second  liquid  phase  is  instable  it 
may  yet  be  formed  temporarily.  When  alcohol  is  added  to  a  strong 
solution  of  sodium  carbonate,  at  ordinary  temperature,  the  liquid 
usually  separates  into  two  layers,  one  of  which  is  instable  and  disap- 
pears in  the  course  of  time.  This  phenomenon  seems  to  be  entirely 
general  and  to  occur  in  all  cases  when  a  solid  is  precipitated  from  a 
solution  by  addition  of  a  liquid  in  which  it  is  not  soluble.  This  was 
first  shown  by  Link  and  by  Schmidt,  and  the  subject  has  since  been 


Zeit.  phys.  Chem.  I,  509  (1887).     *  Ibid.  7,  308  (1891). 


242  The  Phase  Rule 

studied  by  many.1  Frankenheim  found  that  such  salts  as  ammoni- 
um chloride,  magnesium  chromate,  calcium  carbonate  and  the  sul- 
fates  of  sodium,  magnesium,  manganese  and  aluminum  separate  in 
liquid  drops  when  precipitated  by  alcohol.  From  this  it  is  fair  to 
conclude  that  all  salts  are  precipitated  as  an  instable  solution  from 
which  the  solid  then  crystallizes.  This  raises  the  question  whether 
a  solution  of  two  components  may  not  usually  pass  through  the  in- 
stable  state  of  two  solutions  before  crystallizing,  and  this  receives  a 
certain  confirmation  from  the  fact  that  with  salicylic  acid  and  water 
the  curve  for  two  solutions  and  vapor  is  instable  along  its  whole 
length.2  It  is  impossible  to  answer  this  question  directly,  and  so  far 
no  direct  method  of  doing  this  has  been  discovered. 


Ostwald,  L,ehrbuch  I,  1040 ;  Lehmann,  Molekularphysik  I,  730. 
Cf.  also  Ivehmann,  Molekularphysik  I,  726. 


FOUR  COMPONENTS 
CHAPTER  XX 

GENERAL  THEORY 

With  four  components,  six  phases  constitute  a  nouvariant  sys- 
tem, five  a  monovariant  system,  and  four  a  di variant  system.  The 
discussion  will  be  limited  to  cases  in  which  water  is  one  of  the  com- 
ponents, because  no  other  system  has  yet  received  any  attention. 
When  the  other  three  components  are  three  salts,  such  as  the  chlo- 
ride, bromide  and  iodide  of  potassium,  which  do  not  react  with  each 
other  and  which  form  .no  hydrates,  the  only  non  variant  system  pos- 
sible has  the  three  salts  and  ice  as  solid  phases.  In  the  monovariant 
systems  in  which  solution  and  vapor  occur,  the  other  phases  can  be 
the  three  salts  or  any  two  of  the  salts  and  ice.  When  double  salts 
or  hydrates  are  possible,  the  number  of  iionvariant  systems  will  be 
correspondingly  increased.  When  the  constituents  are  water  and 
two  salts,  A  and  B,  which  can  form  two  others,  C  and  D.  by  meta- 
thesis, there  are  four  components.  If  there  is  a  sextuple  point  in 
which  the  four  salts  are  in  equilibrium  with  solution  and  vapor  there 
will  be  two  boundary  curves  for  three  salts,  solution  and  vapor  exist- 
ing at  temperatures  above  the  sextuple  point  and  two  others  exist- 
ing at  temperatures  below  it.  If  the  solid  phase,  along  the  latter  pair 
of  curves  are  A,  B,  C  and  A,  B,  D,  respectively,  the  solid  phases  at 
higher  temperatures  will  be  A,  C,  D  and  B.  C,  D,  the  sextuple  point 
being  an  inversion  point  for  the  two  pairs  of  salt,  A  and  B,  C  and  D. 
When  the  solid  phases  are  the  anhydrous  salts  such  a  sextuple  point 
can  occur  when  the  salts  by  themselves  can  form  the  nonvariant  sys- 
tem, four  salts  and  vapor.  Since  no  instance  of  this  has  yet  been 
found,  no  sextuple  point  of  this  type  has  been  discovered.  If  any 
mixture  of  four  salts,  which  form  no  compounds  with  water,  be  dis- 
solved in  water  and  the  solution  evaporated  to  dry  ness,  one  pair  of 
salts  will  be  found  to  be  stable  and  the  other  pair  instable.  Instances 
of  stable  pairs  of  salts  are  potassium  nitrate  and  sodium  chloride,  so- 


244  The  Phase  Rule 

dium  nitrate  and  Ammonium  chloride.  The  reciprocal  pairs,  potas- 
sium chloride  and  sodium  nitrate,  sodium  chloride  and  ammonium 
nitrate,  are  not  stable  and  will  not  crystallize  from  solution  simulta- 
neously. While  it  is  possible  to  have  potassium  nitrate,  sodium 
chloride  and  sodium  nitrate  or  potassium  nitrate,  sodium  chloride  and 
potassium  chloride  in  equilibrium  with  solution  and  vapor,  it  does 
not  follow  that  this  equilibrium  will  be  reached  unless  three  out  of 
the  four  salts  be  added  to  the  solution.  If  one  starts  with  potassium 
nitrate,  sodium  chloride  and  water,  this  is  really  a  system  containing 
three  components,  because  the  two  salts  do  not  react  and  the  system 
differs  experimentally  in  no  way  from  one  made  up  of  potassium 
nitrate,  potassium  chloride  and  water.  If  we  take  K,  Na,  Cl,  NO3 
and  H2O  as  the  constituents  there  are  two  limiting  conditions  : 

K  =  NO8  and  Na  =  Cl. 

This  is  true  only  so  long  as  the  experiments  are  carried  on  at  tem- 
peratures at  which  potassium  nitrate  and  sodium  chloride  are  the 
stable  pair.  Beyond  the  inversion  temperature  these  two  limiting 
conditions  would  be  replaced  by  the  single  one  : 

K.  +  Na  =  NO9  +  Cl. 

This  can  be  realized  experimentally  by  starting  with  the  instable 
pair  of  salts.  If  we  dissolve  potassium  chloride  and  sodium  nitrate 
in  water  and  evaporate  to  dryness,  the  solid  phases  will  be  potassium 
nitrate,  sodium  chloride  and  either  potassium  chloride  or  sodium 
nitrate,  depending  on  which  salt  is  present  in  excess.  If  the  two 
salts  are  taken  in  equivalent  quantities  there  will  be  no  excess  and 
the  solid  phases  are  potassium  nitrate  and  sodium  chloride.  Under 
these  circumstances  the  system  behaves  as  if  it  contained  only  three 
components  ;  but  this  case  is  not  to  be  confused  with  the  one  where 
the  salt  constituents  are  the  stable  pair.  If  one  starts  with  the  stable 
pair  of  salts  and  water  there  are  three  components  until  the  inversion 
temperature  is  passed  when  the  number  increases  to  four.  If  one 
starts  with  the  instable  pair  and  water,  the  two  salts  being  in  equiv- 
alent quantities,  or  with  the  stable  pair  and  water,  the  two  salts  being 
in  equivalent  quantities,  the  system  contains  three  components 
whether  the  experiments  be  carried  on  above  or  below  the  inversion 


Four  Components  ._• 

temperature.  At  oiie  side  of  the  sextuple  point  the  limiting  condi- 
tions are  : 

K  =  NO,  and  Xa  =  Cl, 

while  on  the  other  side  of  the  point  we  have  : 
K  =  Cl  and  Xa  =  XO,. 

It  is  clear,  of  course,  that  one  could  write  the  limiting  conditions 
somewhat  differently  if  desired  as.  for  instance, 

K  +  Xa  =  XO,  +  Cl  and  K  =  XO, 

K  +  Xa  =  XO,  +  Cl  and  K  =  Cl. 

The  one  set  applies  when  potassium  nitrate  and  sodium  chloride  are 
the  stable  pair  and  the  other  in  the  reverse  case.  This  is  merely  an 
algebraical  transformation  ;  but  it  has  the  advantage  of  simplicity  in 
that  only  one  condition  changes  as  the  system  passes  through  the  in- 
version point. 

When  at  least  one  of  the  salts  contains  water  of  crystallization, 
it  is  not  difficult  to  find  instances  of  sextuple  points  in  which  ice  is 
not  one  of  the  solid  phases.  The  study  of  these  points  has  been  car- 
ried on  entirely  in  van  't  HofFs  laboratory.  At  3.7°  there  can  be  in 
equilibrium  hydrated  sodium  sulfate,  potassium  chloride,  sodium 
chloride,  a  double  sulfate  corresponding  to  the  formula  I^Xaa'SC^  )s, 
solution  and  vapor.  The  first  pair  of  salts  is  stable  below  this  tem- 
perature, the  second  above  it.1  At  10.8°  ammonium  chloride  and  hy- 
drated sodium  sulfate  react  to  form  sodium  chloride  and  sodium  am- 
monium sulfate  with  two  of  water,  the  latter  pair  being  stable  above 
this  temperature.1  At  31°  sodium  chloride  and  hydrated  magnesium 
sulfate  change  into  sodium  magnesium  sulfate  with  four  of  water  and 
hydrated  magnesium  chloride,  the  latter  pair  being  stable  above  this 
temperature.3  By  adding  sodium  chloride  in  excess  the  point  at 
which  the  double  sulfate  of  sodium  and  magnesium  changes  into  the 
single  satiates  is  lowered  to  5°.* 

It  seems  probable  that  in  the  near  future  we  shall  have  complete 
data  for  at  least  one  system  containing  four  components  ;*  but  for  the 


van  't  Hoff  and  Reicher,  ZeiL  phys.  Chem.  3,  482(1889). 
•  van  't  Hoff  and  Tan  Derenter,  Ibid.  I.  165  (1887). 
1  Meverhoffer.  Monatsheft.  Wien.  17,  13  (1896). 


246  The  Phase  Rule 

present  the  on\y  experimental  investigation  is  one  by  Lowenherz.1 
The  system  studied  was  the  one  made  up  of  potassium  chloride,  po- 
tassium sulfate,  magnesium  sulfate,  magnesium  chloride  and  water. 
The  isotherm  for  25°  was  determined,  solution  and  vapor  being 
always  present.  Magnesium  sulfate  and  potassium  chloride  are  the 
stable  pair,  and  if  no  double  salts  were  formed  and  no  new  hydrates 
there  would  be  only  two  monovariant  systems  possible  at  this  tem- 
perature, the  solid  phases  being  the  stable  pair  of  salts  and  potassium 
sulfate  or  magnesium  chloride  with  six  of  water.  It  so  happens  that 
there  are  two  double  salts  with  compositions  corresponding  to  the 
fonnula  K2Mg(SO4)26H2O  and  KMgCl36H2O.  In  addition  we  can 
have  magnesium  sulfate  crystallizing  with  six  of  water  and  the  num- 
ber of  monovariant  systems  becomes  five,  the  solid  phases  being  : 

1.  K2Mg(SO4)26H2O,  KC1,  K2SO4. 

2.  MgSO47H2O,  K2Mg(SO4)26H2O,  KC1. 

3.  MgSO,7H2O,  MgSO46H2O,  KC1. 

4.  MgSO46H2O,  KC1,  KMgCl36H2O. 

5.  MgS046H20,  KMgCl36H20,  MgCl26H3O. 

It  will  clear  matters  up  a  little  to  consider  the  way  in  which  we  can 
pass  from  one  monovariant  system  to  another.  Starting  from  a  solu- 
tion saturated  with  respect  to  potassium  sulfate  and  potassium  chlo- 
ride, the  following  changes  will  take  place  on  adding  anhydrous 
magnesium  chloride  or  on  adding  hydrated  magnesium  chloride  and 
evaporating  off  the  excess  of  wrater.  Magnesium  chloride  and  potas- 
sium sulfate  go  into  solution,  while  potassium  chloride  is  precipitated 
until  the  solution  is  saturated  with  respect  to  the  double  sulfate,  and 
there  are  then  three  salts  in  equilibrium  with  the  solution  ;  potassium 
sulfate,  potassium  chloride,  and  potassium  magnesium  sulfate  with 
six  of  water.  With  further  addition  of  magnesium  chloride,  there  is 
disappearance  of  potassium  sulfate  as  solid  phase  and  precipitation  of 
the  double  salt  and  of  potassium  chloride,  the  concentration  remain- 
ing unchanged  until  the  solid  potassium  sulfate  has  vanished.  The 


1  Zeit.  phys.  Chem.  13,  459  (1894). 


Four  Components 


247 


double  sulfate  will  then  dissolve  and  potassium  chloride  precipitate, 
the  concentration  changing  until  magnesium  sulfate  with  seven  of 
water  appears,  forming  a  new  monovariant  system.  The  double  sul- 
fate then  disappears  with  formation  of  magnesium  sulfate  heptahy- 
drate  and  potassium  chloride.  In  the  next  stage,  addition  of  mag- 
nesium chloride  produces  a  slight  precipitation  of  the  two  salts  just 
mentioned,  the  amount  of  magnesium  chloride  increasing  until  the 
hexahydrate  of  magnesium  sulfate  separates.  The  heptahydrate 
will  next  disappear  and  the  concentration  will  then  change,  the  mag- 
nesium chloride  precipitating  both  the  hexahydrate  and  potassium 
chloride.  The  appearance  of  the  double  chloride  is  followed  by  the 
disappearance  of  the  potassium  chloride  and  then,  with  almost  no 
change  of  concentration,  by  the  appearance  of  crystallized  magnesi- 
um chloride  with  six  of  water,  forming  the  monovariant  system 
marked  V  in  the  list.  The  data  upon  which  these  statements  rest  are 
given  in  Table  XXXI.  Departing  from  the  usual  custom  the  figures 
denote  reacting  weights  of  Kj,  Mg,  C12  and  SO4  in  one  thousand  re- 
acting weights  of  water.  The  figures  in  the  first  column  show  the 
monovariant  systems  to  which  the  concentrations  refer. 

TABLE  XXXI 


K2 

Mg 

a, 

SO, 

I  . 

25 

"*2 

46 

1  1 

2 

71 

64 

16 

3  • 
4  • 
5  • 

....     8 

....      2 
....      2 

77 
no 
III 

70 

100 
IOI 

15 

12 
12 

It  is  to  be  noticed  that  the  two  magnesium  sulfates  with  seven 
and  with  six  of  water,  can  exist  simultaneously  in  equilibrium  with 
solution  and  vapor  at  a  series  of  concentrations  for  each  temperature. 
If  we  are  to  consider  the  appearance  of  the  second  salt  as  due  to  de- 
hydration, the  part  of  the  isotherm  along  which  these  two  hydrates 
are  the  solid  phases  must  also  be  an  isobar  or  line  of  constant  pres- 


248  The  Phase  Rule 

sure.  There  is  nothing  in  the  solubility  determinations  to  vitiate 
this  conclusion.  L,6wenherz  finds  that  the  two  hydrates  are  in  equi- 
librium with  a  solution  containing  fifteen  reacting  weights  of  magne- 
sium sulfate  and  seventy-three  reacting  weights  of  magne  ium  chlo- 
ride in  one  thousand  reacting  weights  of  water,  while  the  concentra- 
tion of  the  solution"  in  equilibrium  with  the  two  hydrates  and  potas- 
sium chloride  is  given  under  3  in  Table  XXXI.  Assuming  complete 
electrolytic  dissociation  there  would  be  two  hundred  and  forty-nine 
units  in  the  first  solution  and  two  hundred  and  forty-eight  in  the  sec- 
ond, numbers  which  are  identical.  This  does  not  prove  anything 
because  the  assumption  is  not  fulfilled,  and  it  is  impossible  to  replace 
it  by  one  which  certainly  represents  the  facts.  Since  it  is  not  im- 
probable that  the  degree  of  dissociation  is  much  the  same  in  the  two 
solutions,  this  coincidence  is  sufficient  to  make  an  experimental  study 
of  this  question  very  much  to  be  desired. 


ERRATA. 

On  page  19  it  is  stated  that  "  at  constant  pressure  the  addition  of 
heat  produces  an  increase  of  volume  "  which  is  not  true  for  liquid 
water  between  o°  and  4°.  In  the  discussion  on  pages  10  and  26,  it 
should  therefore  have  been  pointed  out  that  the  monovariant  system, 
water  and  water  vapor,  will  change,  if  cooled  at  constant  volume, 
into  the  di variant  system,  liquid  water,  provided  the  ratio  of  the 
volumes  of  liquid  and  vapor  be  very  large  at  4°. 

On  page  30  strike  out  from  "  Although  these  results"  to  the  end 
of  the  paragraph. 

On  page  44  I  had  forgotten  a  second  paper  by  Hannay,  Proc. 
Roy.  Soc.  30,  484  (1880),  in  which  he  used  a  saturated  solution. 

In  Fig.  5  the  dotted  lines  should  not  appear  to  pass  through  a 
minimum  pressure  before  meeting  OB. 

On  page  177,  instead  of  "  To  obtain  the  salt  pure  .  .  .  ,"  read  : 
The  crystals  will  contain  potassium  chloride  as  impurity  if  washed 
with  water  or  a  potassium  chloride  solution  and  copper  chloride  as 
impurity  if  washed  with  a  copper  chloride  solution. 


INDEX    OF    AUTHORS 


Alexejew,  69,  102,  103,  106,  127 
Alkemade,  van,  69,  147,  149,  166,  222. 
Altschul,  14. 
Amagat,  18,  20. 
Ambronn,  39. 
Andrews,  14. 
Appleyard,  200. 
Arctowski,  49. 

Babo,  v.,  44. 

Bailey,  50,  92. 

Bancroft,  35,  36,  39,  43,  47,  69,  73,  81,  92,  98, 

99,  101,  103,  127,  147,  159,  180,  240. 
Bathrick,  159. 
Battelli,  18. 
Barus,  n. 
Bauer,  A.,  124. 
Bauer.  A.  E.,  134. 
Beckmann,  136. 
Beilstein,  124. 
Bemmelen,  van,  200. 
Berthelot,  D.,  31,  34. 
Berthelot,  M.,  41,  239. 
Bijlert,  van,  136,  144. 
Le  Blanc,  39,  202. 
Bodlander,  203,  241. 
Boguski.  29. 
Bois-Reymond,  du,  138. 
Bosse,  212. 
Braun.  4,  51,  52,  54. 
Brodie,  29. 
Brown,  118. 
Budde,  82. 

Cailletet,  14,  15,  25. 

Carnelly,  42,  134. 

Chancel,  41. 

Chappuis,  140. 

Le  Chatelier,  i,  4,  30,  35,  41,  65,  76,  199. 

Cohen.  57. 

Colardeau,  14,  25. 

Colson,  36,  52. 

Coppet,  de,  175. 

Dahms,  129. 

Dalton,  35. 

Damien,  18. 

Damnier,  31,  59. 

Debray,  65. 

Demarcay,  16. 

Deventer,  van,  42,  180,  181,  189,  190,  245. 

Deville,  139. 

Dewar,  iS. 

Dieterici,  44. 

Ditte,  178. 

Donny,  23. 

Dufour.  23. 


Duhem,  4,  22. 

Emden,  44. 

Etard,  41,  42,  44,  48,  49,  51,  128,  133,  134,  158. 

Faraday,  15. 

Favre,  52. 

Ferche,  25. 

Ferratini,  136. 

Fock,  35,  36,  199,  203,  210,  212. 

Frankenheim,  242. 

Galitzine,  14,  35.        • 

Garelli,  136, 

Gautier,  143,  144,  145. 

Gernez,  23,  32,  68. 

Gibbs,  i,  2,  4,  22,  69,  100,  122,  147,  226,  234. 

Gooch,  92. 

Goldschmidt,  181,  187. 

Gossens,  6. 

Graham,  200. 

Guldberg,  45. 

Guthrie,  38,  40,  45,  48,  103,  107,  117,  125,  129, 

133,  142,  153,  154,  157,  175- 
Guye,  119.     - 

Hagen,  143. 

Hallock,  16. 

Hannay,  36,  44,  92. 

Hautefeuille,  33. 

Haywood,  98. 

Heide,  van  der,  165,  167,  170,  180,  216. 

Heilborn,  15. 

Helmholtz,  H.  v.,  139. 

Helmholtz,  R.  v.,  22. 

Heycock,  142,  144. 

Hoff,  van' t,  3,  29,  35,  136,  139,  172,  173,  180,  181, 

189,  190,  232,  245. 
Hoitsema,  140,  232. 
Horstmann,  127. 

Isambert  65,  198. 

Jakowkin,  239. 
Joannis,  6b,  65. 
Jorissen,  181. 
Jungfleisch,  239. 

Konowalow,  59,  96,  98,  99,  100,  101,  118,  119. 

Kopp,  142. 

Kuntze,  199,  203,  310. 

Kiister,  39,  136,  137,  144.  199,  200. 

Landolt  and  Bornstein,  14,  40,  48,  154. 

Lang,  v.,  20. 

Lehfeldt,  125. 

Lehmann,  23,  32,  33,  34,  93,  129.  133,  134.  137. 

138,  143,  242. 
Lescoeur,  60. 


252 


Index  of  Authors 


Linebarger,  85,  119,  125. 

Link,  241. 

Loweuherz,  186,  246,  248. 

MacGregor,  52. 

Mallard,  30. 

Marsden,  36. 

Margueritte-Delacharlonna>,  92. 

Masson,  103. 

Mclntosh,  237. 

Meslans,  31,  33,  140,  145. 

Meyerhoffer,  22,  33,  43,  146,  169,  170,  174,  175, 

177,  178,  180,  181,  206,  209,  213,  215,  216,  231, 

245- 

Miolati,  116,  129. 
Mitscherlich,  31.  • 

Mond,  140. 
Morse,  92. 
Moser,  23, 
Muthmann,  199,  203,  210. 

Nadeshdin,  14. 

Natterer,  15. 

Nernst,  2,  5,  21,  27,  35,  46,  92,  93,  100,  103,  108, 

234,  239- 
Neuberg,  241. 
Neville,  142,  144. 
Nicol,  158,  159,  202,  203. 
Noyes,  202. 

Oersted,  20. 

Offer,  38. 

Olszewski,  15, 

Orndorff,  124. 

Ostwald,  18,  20,  23,  35,  42,  44,  51,  52,  58,  63,  68, 

92,  97,  98,  99,  122,  124,  127, 129,  130,  138,  142, 

166,  200,  201,  235,  242. 

Pagliani,  20. 

Parraentier,  41. 

Paternd,  136.  4 

Pfauiidler,  38. 

Pfeffer,  235,  236. 

Pfeiffer,  240. 

Pickering,  107,  113,  114,  131,  132,  220. 

Pictet,  15. 

Prendel,  28. 

Prytz,  91. 

Ramsay,  12,  17,  25,  140. 

Raoult,  44,  236. 

Reicher,  29,  30,  31,  32,  42,  91,  179,  180,  181,  245. 

Remsen,  38. 

Retgers,  207. 

Riecke,  31. 

Roberts-Austen,  36,  162. 

Roloff,  50,  129. 

Rontgen,  20. 

Rose,  33. 

Roscoe,  122. 


Roozeboom,  3,  24,  30,  31,  35,  41,  42,  43,  61,  63, 

65,  69,  71,  75,  76,  78,  80,  81,  82,  83,  85,  91,  98, 
106,  107,  109,  in,  112,  113,  114,  137,  140,  147, 
M9.  I52,  153,  155,  162,  163,  168,  169,  172,  174, 
179,  188,  189,  190,  199,  201,  204,  205,  206,  208, 

209,  212,  217,  219,   221,   222,  223,  226,   231,   232. 

Ruys,  31. 

Sajoiitchewsky,  14. 

Schmidt,  C.,  241. 

Schmidt,  G.  C.,  200. 

Schneider,  20,  36. 

Schreiiiemakers,   146,   147,   150,   159,  166,   168, 

178,  183,  196,  201,  202,  205,  213,  214,  219,  223. 
Schroder,  23. 
Schrotter,  23,  33. 
Schumann,  20. 
Schultz,  38,  130,  143. 
Schiitzenberger,  93. 
Schwarz,  34,  133. 
Shields,  140. 
Sheiistone,  41,  129. 
Spring,  16,  21,  133,  189,  232. 
Stackelberg,  v.,  51. 
Stock,  136. 
Stokes,  147. 
Stortenbeker,  35,  86,  89,  199,  201,  204,  207,  209, 

212. 
Story-Maskelyne,  36. 

Tammaun,  142,  144. 

Tegetmeier,  36. 

Thonia,  139. 

Thomson,  18.  • 

Thorpe,  118. 

Thurston,  147. 

Tilden,  41,  129. 

Traube,  ].,  241. 

Traube,  M.,  235. 

Trevor,  4,  100,  205,  212,  226. 

Troost,  33. 

Valson,  52. 
Vaubel,  42. 
Vicentini,  20. 
VignoiVi29. 
Villard,  92. 
Violle,  36. 
Voigt,  18. 
Vriens,  194. 

Waals,  van  der,  14,  23,  89. 
Wald,  25,  226. 
Walden,  235. 
Walker,  96,  200. 
Warburg,  36. 
White,  92. 

Wiiikelmann,  45,  125. 
Wroblewski,  15. 
Young,  12,  17,  25. 


INDEX 


Absorption.  138.  Crit 

Adsorption,  138,  200.  temperature,  14.  iS,  44. 

Acid,  benzoic,  105.  Cryohydrate,  39. 

formic,  118.  122.  Cryohydric  temperatnre,   39,   117,   139.  157. 

hydrobromic,  91,98,  112.  118,  122.  178. 

hydrochloric,  91,  114,  118,  122,  229.  Curve,  boundary.  25,  149. 

salicylic.  105.  242.  fusion,  45.  91,  94,  96.  106.  117,  izS,  132. 

titanic.  33.  middle,  152. 

Alcohol,  butyl,  103.  side,  152. 

ethyl.  45.  127,  159.  240,  241.  solubility.  45.  104.  105,  187. 

isobutyl,  103.  sublimation.  25. 

methyl,  92,  127,  236.  vaporization,  24. 

propyl,  118,  121,  123,  127. 

Allotropic  modifications.  A  31.  33.  34.   71,  Deliquescence,  51,  64- 

87,  93.  "9,  133.  MS-  Densit>r  «»*>•*  7- 

Alloy,  142.  Diagrams,  24,66.  69.  99.  123.  146.  159.  I7o.  202. 

eutectic,  117,  129,  M9.  156-  ^  **1- 

Ammonia,  and  ammonium  bromide,  -ft.  concentration-pressure,  96. 118,  uB. 

of  crystallization,  6s  concentration-temperature,  66,  70.  75,  79. 

and  sodium,  61.  65.  *r  «L  94,  101.  105,  106.  no,  116,  xao. 

Ammonium  chloride,  17,  181,  198,  201.  206.  **•  '31*  »3i  «35. 

212,  217.  227,  242,  244.  245.  pressure-temperature,  24,  28.  32,  38.  46. 
Antimony,  144.  M5-  57,  7L  72.  73.  «7.  «*,  112,  123.  184.  186. 
Arragonite,  stability  of,  53.  l8^  19^.  192- 

triangular.  148,  154,  157,  161,  165,  172,  176. 

Barium  carbonate,  dissociation  of,  65.  193.  220. 

Boiling  point,  12.  Diaphragm,  semipermeable,  235,  236,  237. 

change  with  pressure,  13.  Diethylamine,  103,  132. 

of  d  ivariant  systems,  50,  101,  110.  Dissociation,  65.  187,  189.  197. 

of  monovariant  systems,  12,  43,  go,  87.  95,  Distillation,  of  consolnte  liquids,  120 
101.  of  partially  miscible  liquids,  102. 

Break.  67,  126,  132,  152,  139,  169.  240.  Dfrariant.  3. 

systems.  3,  4,  37.  146.  243. 

Calcite,  stability  of,  33.  , 

Calcium  carbonate,  dissociation  of,  65.  Efflorescence,  59.  62,  64,  iSS,  193. 

Equilibrium,  complete  heterogeneous.  3. 

butyrate,  41.  incomplete  heterogeneous,  v 

isobutyrate,  41.  labile,  22. 

chloride,  71-  stable.  22. 

snlfate,  41.  Eutectic,  alloy,  117,  i^,  149.  15^ 
Capillary  tenskm.  2,  8.  temperature,  117,  129. 

Carbon,  stability  of  modifications  of,  33. 

Chlorine,  86, 112,  Ferric  chloride,  78.  201,  206,  217.  219, 

Component,  I,  226,  227,  234.  Formula,  use  of,  229. 

Compound,  I,  35,  143.  Fractional,  crystallization,  2i& 

chemical  and  molecular,  56.  distillation,  102.  120.  124. 

Consolnte,  35.  evaporation,  309. 

temperatnre,  103.  127,  143.  Freedom,  degrees  of,  3,  230^ 

Constituent,  226,  227.  Freezing  mixtures,  47. 

Copper,  acetate,  179,  181,  188.  Freezing  point,  7. 

chloride.  175,  180,  181,  191,  194,  201,  212,     '  change  with  pressure.  18,  49-  9i> 

213.  215,  216.  of  solutions,  46,  142, 
snlfate,  aoi,  202,  204.  205.  207,  208.  212,  230.  of  solid  solutions,  13&  144- 


254 


Index 


Fusion  curve,  45,  91,  94,  96,  106,  117,  128,  132. 
Fusion  point  of  two  solid  phases,  163,  222. 

Gravity,  effect  of,  2,  5,  13,  20,  100. 
Heat,  addition  and  subtraction  of,  8,  12,  16, 
17,  19,  27.  54.  232. 

relations,  7. 

reservoir,  9. 

of  solution,  42,  102,  103. 
Hydrate,  56. 

melting  point  of,  64,  72,  79,  112,  131,  219, 
221.' 

vapor  pressure  of,  59,  73,  81,  186,  189,  194. 

Inversion,  point,  25,  57,  75,  93,  180,  233,  239, 

243- 

pressure,  3,  6,  25,  30. 
temperature,  3,  6,  25,  29,  30,  33,  61,  133, 

180,  238,  239. 
Iodine,  86,  92,  136,  199. 

Isotherm,  155,  156,  161,^201,  209,  213,  223,  240, 
241,  246. 

Labile,  22. 

Lead  iodide,  179,  195,  205,  213. 

nitrate,  153,  202. 
Liquefaction  of  gases,  15. 
Liquids,  properties  of,  20. 

supercooled,  23. 

superheated,  24. 

Magnesium,  chloride,  245,  246. 

sulfate,  165,  172,  180,  190,  191,  195,  201,  202, 
205,  207,  212,  213,  216,  240,  242,  245,  246. 
Manganous,  chloride,  212.      « 

sulfate,  85,  208,  240.  242. 
Matter,  existence  of,  5. 
Mercuric  iodide,  134,  151. 

sulfate,  230,  232. 
Metals,  142. 

diffusion  of,  21. 

volatility  of,  16. 
Monovariant,  3. 

systems,  3,  4,  37,  146,  235,  237. 

Naphthalene,  6,  18,  25,  44,  94,  104,  116. 
Nonvariant^3. 

systems,  3,  4,  37,  146,  235,  237,  243. 

Occlusion,  138. 

Palladium  and  hydrogen,  139. 

Pattinson  process,  142. 

Phase,  i. 

Phenanthrene,  116. 

Phenol,  103,  105,  136. 

Phosphorus,  32. 

Potassium,  chlorate,  133,   137,  201,  204,  207, 

212. 

chloride,  39,  157,  175,  180,  i9I,  194,  201,  212, 
213,  215,  216,  228,  231,  244,  245,  246. 


Potassium,  hydroxide,  131,  227,  240. 
iodide,  44,  92,  179,  195,  205,  213. 
nitrate,  129,  133,  153,  157,  201,  202,  207,  227, 

228,  231,  243,  244. 
perchlorate,  201,  203,  210. 
permanganate,  201,  203,  210. 
sulfate,  165,  180,  191,  201,  203,  205,  207, 

210,  212,  216,  230,  244. 
Potential,  chemical,  2. 
Pressure,  and  boiling  point,  13. 
critical,  14. 

and  freezing  point,  18,  49,  91. 
inversion,  3,  6,  25,  30. 
osmotic,  235,  236. 
and  solubility,  51. 

vapor,  11,  16,  43,  50,  59,  73,  81,  87,  91,  96,  99, 
118,  185,  186,  189,  194,  198,  248. 

Racemates,  181,  233. 

Range  of  decomposition,  174. 

Reaction,  chemical  and  physical,  5. 

velocity  of,  25,  31,  32,  85. 
Resistance,  passive,  226,  234. 

Silver,  iodide,  134,  151. 

nitrate,  128,  133,  201,  207. 
Sodium,  and  ammonia,  61,  65. 

chloride,  54,    159,    184,   201,   210,   228,   231, 
243,  244,  245. 

nitrate,  128,  153,  201,  202,  210,  231,  244. 

sulfate,  41,  56,  172,  180,  184,  190,  195,  201, 

205,  207,  212,  213,  232,  240,  242,  245. 
Solids,  properties  of,  21. 

volatility  of,  92. 
Solubility,  change  wilh  pressure,  51. 

change  with  temperature,  41. 

and  chemical  nature,  42 

of  consolute  liquids,  127,  240. 

curve,  45,  104,  105,  187. 
Solute,  35,  42,  94,  109,  147,  154,  157. 

change  of  solute,  89,  106,  128,  133. 
Solution,  35. 

heat  of,  42,  102,  183. 

solid,  35,  135,  144,  199,  201,  204,  239. 

three  liquid,  239. 

two  liquid,  94,  96,  99,  101,  103,  107,  112,  238, 
240,  241,  242. 

supersaturated,  68,  81,  221. 

vapor  pressure  of,  43,  50,  87,  91,  96,  99, 

118,  185,  194. 
Solvent,  35,  43,  94,  109,  147,  154,  157,  183. 

change  of,  89,  106,  128,   133. 
Sublimation,  curve,  25,  186. 

point,  17. 
Sulfur,  28. 

and  toluene,  44,  93,  98,  103. 

and  xylene,  98,  102. 
Sulfur  dioxide,  98,  107. 


Index 


255 


Tartrates,  181,  233.  Vapor,  pressure,  n,  16,  198. 

Temperature,  consolute.  103,  127,  143.  pressures  of  hydrates,  59.  73,  81,  186,  189. 

critical,  14,  18,  44.  194,  248. 

cryohydric.  39,  117,  129,  157,  178.  pressure  of  solutions,  43, 50,  87,  91, 96, 99, 

eutectic,  117,  129.  118,  185,  194. 

inverskm,  3,  6,  25,  29.  30,  33,  61,  133,  180,  supercooled,  7,  22. 

238,  239.  Vaporization  curve,  24. 

Thallium  chlorate,  157,  201,  204,  212.  Variable,  dependent,  226. 
Thorium  sulfate,  41,  85.  independent,  2,  226,  235. 

Toluene,  93.  98,  103.  Volatility,  of  metals,  16 
Transformation  point  for  two  solid  phases,  of  solids,  92. 

2*3- 

Triethylatnine,  106.  Water.  16.  24,  44. 

of  crystallization,  56,  201. 

Vapor,  and  gas,  15.  Work,  addition  and  subtraction  of,  8,  n,  16, 
properties  of,  19.  19,  27,  53. 


D    001  068  334 


cun 

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